Variations in gravity due to height and topography

Topography influences gravity measurements because it causes variations in station elevation, i.e. the distance between the station and the centre of the Earth’s mass, cf.

Eq. (3.3). Also significant are the effects of the materials

forming the mass of the topography. They exert their own gravitational attraction which tends to oppose the decrease in gravity caused by increasing elevation, but as predicted byEq. (3.3), the attraction of the mass has less effect than the distance factor. Terrain effects can be as complicated as the terrain itself; they are particularly strong in gravity gradient measurements.

Compensation of height and topographic effects involves three sequential corrections, known as the free- air, Bouguer and terrain corrections. When the gravity meter is located on the surface of the topography, the usual situation for a ground survey, all three corrections are then based on the height of the topography above some datum

90°N 60°N 60°S 90°S 30°N 30°S 0°

180°W 120°W 60°W 0° 60°E 120°E 180°E

a)

b)

Geoid

Reference spheroid

Topographic surface Vertical – the direction along which gravity is measured

Positive separation Negative separation –30 0 –30 30 60 30 –30 30 –60 60 –90 0 0 0 30 30 –30 –30 –60 0 –60 –30

Figure 3.15 (a) Relationship between the undulating geoid and the smooth reference spheroid. Vertical is everywhere perpendicular to

the geoid. (b) Separation between the WGS84 spheroid and the EGM96 geoid. Computed using data from Lemoine et al. (1998).

level, usually the geoid (sea level), so they can be combined into a single elevation correction. When the gravity meter is elevated above the topography (Fig. 3.16a), e.g. on a tripod or in an aircraft, it is not possible to apply a single height- dependent elevation correction. Usually the three correc- tions are applied separately and in the sequence described below.

A common misconception regarding these corrections is that their application produces the equivalent reading that would have been obtained if the gravity meter had been located at the datum level. This is not the case, since such a process must include continuation of thefield (seeSection 3.7.3.2). Instead, what the reduction process does is correct for height and topographic effects whilst retaining relative changes due to lateral density changes in the subsurface.

Another influence related to topography is the isostatic state of the survey area, in particular whether local or regional isostatic compensation occurs (Lowrie, 2007). For example, the mass deficit of a mountain’s root in local Airy-type compensation leads to lower gravity than a set- ting where theflexural rigidity is sufficient to carry the load

of the mountain in a Pratt-type model. This is because the mass deficiency comprising the root zone partly balances the gravitational consequences of the mass excess repre- sented by positive topography. If there is no local compen- sation, then obviously the gravitational effects of the terrain alone affect the reading. The isostasy-related changes in gravity tend to be of sufficiently long wave- length that they appear as a constant component in most exploration-related gravity surveys. They can be compen- sated using an isostatic correction (Lowrie,2007), import- ant for very large regional surveys, but not normally required for smaller mineral exploration-scale surveys. 3.4.5.1 Free-air correction

If a measurement of gravity were made onflat ground and then, at the same location, the measurement repeated at the top of a tall step ladder, the value of observed gravity would be less at the top of the ladder owing only to the increase in distance between the measurement location and the Earth’s centre of mass, cf. the 1/r2_{component of}

Eq. (3.3). The change in gravity with height is compensated

with the free-air correction (gFA) given by:

g_FA¼2G

R h ð3:15Þ

where h is the height (m) of the gravity station above the datum (usually the geoid), R is the average radius (m) of the Earth (6371 km) and G the universal gravitational constant given inSection 3.2.1.1. This reduces to:

g_FA¼ 3:086h gu ð3:16Þ

which is simply the free-air gradient multiplied by the height. Station height needs to be measured with an accur- acy of about 3 cm in order to calculate the free-air correc- tion to an accuracy of 0.1 gu. The free-air correction compensates for height variations of the gravity meter above or below the datum level, where the station height is that of the topography (htop) plus the height above the

ground surface of the gravity meter (hclr) (Fig. 3.16a). The

free-air correction is added to the gravity reading, noting that height is negative for stations below the datum level so the free-air correction is then negative.

3.4.5.2 Bouguer correction

The free-air correction only accounts for the difference in height between the instrument and the datum level. When the height variation is caused by topographic variations, as is usually the case, the intervening mass of a hill above the a) Datum level Height due to topography (htop) Height due to topography (h_top) Height due to ground clearance (h_clr) b) A B c) Gravity meter Datum level A⬘ Slab Datum level

Figure 3.16(a) The relationship between instrument height, topography and the datum level for the free-air correction. (b) The concept of the Bouguer slab. The significance of the various hatched areas is described in the text. (c) Representation of topographic

variation (in 2D) by a series offlat-topped prisms. Note the smaller

size of the prims close to the gravity station.

datum, or lack of it in the case of the valley below the datum, affects gravity. When the gravity station is located on topography above the survey datum, it is located on a layer of rock that contributes mass between the gravity meter and the datum. The rock layer increases the meas- ured value of gravity. Its effect is compensated by the Bouguer correction (gBOU), which assumes that the Earth

isflat, i.e. not curved, and that the rock layer is a horizontal slab of uniform density extending to infinity in all direc- tions, i.e. the hatched area comprising zones A and A0 in

Fig. 3.16b. It is given by:

g_BOU¼ 2πGρhtop ð3:17Þ

where htop is the height (m) of the topography, or the

thickness of the slab (m);ρ is the Bouguer density (g/cm3), the average density of the slab between the gravity station and the datum level; and G the universal gravitational constant given inSection 3.2.1.1. This reduces to:

g_BOU¼ 0:4192ρhtopgu ð3:18Þ

The Bouguer correction is subtracted from the free-air gravity, noting that height is negative for stations below the datum level so the Bouguer correction is then negative. The Bouguer correction is an approximation as it does not account for variable topography around the station (the top of the slab is flat). Furthermore, selecting the appropriate Bouguer density can be a problem. The aver- age crustal density (2.67 g/cm3; see Section 3.8) is often used but other values are appropriate for particular geological environments. For example, values as low as 1.8 g/cm3may be used in sedimentary basins, 0.917 g/cm3 for ice-covered areas.

Since it is unlikely that the density of the rocks under- lying the entire survey area will be the same, a variable Bouguer density may be more appropriate. The subsurface geology and density distribution needs to be known to determine the appropriate Bouguer density which, obvi- ously, is a problem since determining subsurface density variations is the aim of the gravity survey itself (an example of the geophysical paradox; cf. Section 1.3). If variable density is chosen based on outcrop geology, an explicit assumption is being made that the geology is consistent vertically downwards to the datum level, which may well not be the case. Selecting a single density is usually the only practical solution, reflecting an acceptance of one’s ignor- ance. The resulting errors are small, except in rugged terrain. If variable density is chosen, the datum level should be as high as possible, for example the lowest topographic

level in the survey area, as this reduces the thickness of the slab and will reduce errors due to variations in density between the surface and the datum. The use of variable or constant Bouguer density and the consequences thereof are described in detail by Vajk (1956).

Using the average crustal density of 2.67 g/cm3, the Bouguer correction is 1.12 gu/m, so station height needs to be known with an accuracy of about 10 cm in order to calculate the Bouguer correction to an accuracy of 0.1 gu. Note that the likelihood of non-uniform density is not accounted for in this calculation.

The infinite flat-slab model of the Bouguer correction does not account for the curvature of the Earth’s surface. This can be included in the reduction process by using spherical cap correction; see LaFehr (1991). The spherical cap correction is a more appropriate model in areas of rugged terrain, particularly where there are very large changes in station height.

3.4.5.3 Terrain correction

The Bouguer correction assumes that the rock occupying the height interval between the datum level and the station is a uniform slab extending to infinity in all directions. Referring toFig. 3.16b, the Bouguer correction removes the effect of the mass in zones A and A0. It fails to account for mass above the slab (B), i.e. mass above the gravity station. In contrast, the correction accounts for too much mass in regions where the topographic surface is below than the station, i.e. the non-existent mass where the slab is above the actual ground surface (A0). The terrain correction explicitly addresses these limitations and therefore must be used in conjunction with the Bouguer correction.

Referring to Fig. 3.16, the vertical component of the gravitational attraction of the mass above the slab (in zone B) acts against the gravitational attraction of the subsurface to‘pull’ the gravity sensor up. The observed gravity is then ‘too low’ and a positive correction is required. In the lower- lying area, zone A0, the Bouguer correction has assumed that mass is present here and in so doing has over-corrected; it has taken mass away, reducing the gravitational attraction. Since the Bouguer correction is subtracted, the corrected gravity is too small and again a positive correction compen- sates for this.

The terrain correction is the gravitational attraction, at the gravity station, of all the hills above the Bouguer slab and all the valleys occupied by the slab. It is obtained by determining the mass of the hills and the mass deficiencies of the valleys using topographic information and the

Bouguer density. The process is repeated for each gravity station as they will all (or mostly) have different heights so the Bouguer slab at each station has different thickness (Eq. (3.17)), and they all have a different relationship with the topography. Alternatively, the Bouguer correction can be ignored, and the gravitational attraction of the undulat- ing terrain surrounding the gravity station can be computed as a full terrain correction with respect to the datum level.

The gravitational attraction of topography depends on the size of the topographic features, and decreases with their increasing distance from the gravity station. Depending on the desired accuracy of the survey and the ruggedness of the terrain, this means that relatively small features close to the survey station, such as culverts, storage tanks, reservoirs, mine dumps, open-pits and rock tors, can have a significant effect on the measured gravity and can be a major source of error in high-resolution gravity work

(Fig. 3.17; Leaman, 1998). In addition, topographic fea-

tures tens of kilometres away from the station, and even very large mountain ranges more than a hundred kilo- metres away, may need to be accounted for. The effects of more distant features appear as a regional gradient in the data so their effect could be removed with the regionalfield (seeSection 2.9.2). The terrain correction is by far the most

complex correction to implement as it requires the topog- raphy and its density distribution be accurately known; but this is usually difficult to achieve, so it is prone to error.

Digital terrain data provide the essential heights and shape of the topography needed for calculating the terrain correction. The ability to mathematically describe the ter- rain in terms of the digital data depends on the nature of the data and its resolution. Some airborne gravity systems are equipped with a laser scanner which maps the terrain in an across-line swathe below the aircraft, providing ter- rain information of sufficient resolution and accuracy for the terrain correction. For ground surveys, data may be true point (spot) heights taken from aerial photography or contour maps, or may be average heights of compartments subdividing the photography or contour map. It is import- ant that the actual elevations of the gravity stations match the equivalent points on the DEM as discrepancies in heights and locations are sources of errors. The DEM may lack sufficient resolution to adequately define small topographic features and large abrupt surface irregularities, such as the edges of steep cliffs and the bases of steep hills, in the immediate vicinity of the gravity station, an additional source of error for stations affected in this way. During the gravity survey it is necessary to record details of small local features manually. Leaman (1998) provides practical advice regarding terrain effects close to the gravity station.

Various procedures for geometrically describing the digital terrain, so that its volume above the datum level can be calculated, have been implemented. The most common approach divides the terrain into a large number of volume elements (voxels), usually flat-top juxtaposed prisms, extending down to the datum level. Their gravita- tional attractions are computed and summed at each grav- ity station (Fig. 3.16c) and the whole process repeated for each gravity station. Given that topography closer to the gravity station exerts greater influence than more distant terrain features, efficient algorithms implement smaller prisms with more realistic, i.e. topography resembling, upper surfaces close to the station in order to minimise errors. Distant features, having a smaller gravity effect, are described more crudely with larger and fewer voxels, which also significantly reduces computing resources. Specific densities can be assigned to each prism but, as with the Bouguer density, using variable density implies significant understanding of the density variation in the survey area and its (distant) surrounds.

For gravity stations located near or over a mass of water, such as a lake, the terrain correction needs to correct down

Metres 0 0.1 0.2 0.5 0.1 0.6 0.6 0.2 0.5 0.1 0.2 0.1 0.3 0.1 0.3 0.7 0.1 0.5 1.0 0.1 0.2 0.7 1.3 0.7 0.2 0.1 1.3 0.6 2

Figure 3.17Estimated magnitude of the terrain corrections, in gu, due to small-scale topographic features close to the gravity station.

Redrawn, with permission, from Leaman (1998).

to thefloor of the lake. Also, glaciers (density ranging from 0.79 to 0.88 g/cm3) need to be accounted for in the terrain correction. A summary of terrain corrections, including unusual situations, is provided by Nowell (1999).