The free-air correction

The free-air correction is applied to compensate for variations in the distance of gravity stations from the Earth's centre; in common practice sea level is considered as the reference surface approximating to the geoid (the equipotential surface at mean sea level) and the height of the station above or below this level is used in the calculations. The effects o f topographic masses are ignored and individual free-air anomalies are therefore strongly dependent on elevation. However, if the topography is isostatically

compensated, i.e. if there is no net mass excess or deficiency in the Earth's uppermost

layers, the integral o f free-air anomaly over a sufficiently large area should be zero. Non-zero average o f free-air anomalies therefore indicate departures from local isostatioc equilibrium.

The derivation of free-air correction, is given by Heiskanen and Vening Meinesz (1958, p. 148), who show that

g F = 2 ^ h { \ - ~ - ) 3.7

r 2r

where gm is the force of gravity midway between the point of measurement and sea

level and r is the local radius of the earth. The first term in this equation amounts to about 0.3086 m G al.m '\ The second term in the Equation 3.7 is almost negligible being only about 0.07 mGal.m'^ at 1000 metres altitude but at altitudes over 2000 metres the effect becomes significant. For instance, at 2000 metres the second term amounts to approximately 0.3 mGal.m'^ and at 5000 metres it reaches about 1.7 m Gal.m '\

A less often considered question concerns the application o f the terrain

corrections to free-air anomalies and the related difficulties introduced by local changes in earth curvature (Milsom 1971). Whether or not a terrain correction should be applied depends entirely on the definition o f free-air anomaly adopted, but since anomaly maps are normally used to investigate the state of stress in the earth, it may be thought desirable that the integral over a wide area in which isostatic equilibrium prevails should be zero. St. John (1967) pointed out that corrections should, strictly speaking, be applied for the effects o f topographic masses above the level o f the station, as these masses lie outside the closed surface to which Green’s theorem may be applied. In addition such masses may be compensated isostatically by mass deficiencies at depth. In this case both the excess mass (above the observation point) and the deficit mass (below the observation point) affect the gravity field at the station in the same sense and the combined effect may be significant. Terrain corrections for stations situated on mountain peaks and broad valleys may be negligible (St. John 1967). Valleys below the level of the station do not necessitate a terrain correction to the free-air anomaly on any system since both the mass deficit and any compensation at depth lie within the Green’s theorem surface. In cases where a station is located at the bottom o f narrow river valleys (Milsom 1971), a terrain correction has sometimes been applied by assuming an Airy type isostatic compensation of the external topography. The effect of the

correction is to increase the value of the free-air anomaly computed for low-lying stations in mountainous areas, and therefore to reduce to some extent the very steep gradients normally associated with the unsmoothed anomalies. The correction appears valid in terms o f the requirement o f zero free-air integral in stable areas, but individual values may differ significantly from those of free-air anomaly as normally obtained. In this thesis the more common definition o f free-air anomaly is adopted and terrain corrections have not been applied.

Errors in free-air anomaly arises from inaccuracies in the measurement o f both gravity field and station height; as discussed in the previous chapter, the errors in gravity measurements in the Sorong Fault Zone Project were small and can therefore usually be ignored in comparison with errors due to inaccurate elevations. The probable error in elevation o f stations situated close to sea level is o f the order of two metres, corresponding to approximately 0.62 mGal in free-air correction.

Even if a region is in a state of complete isostatic equilibrium, the gravitational effects o f the topographic masses and their compensation at depth do not cancel for the surface observations, although this condition may be approached in extensive plateau areas. The inverse square law and directional effects ensure that the field due to large surface masses is dominant near the gravity station and that the field due to the

compensation becomes relatively more important as distance increases. In this respect the most effective part of the topography is that which lies directly between the geoid and the gravity station. The higher the station, the greater the mass involved, so that free-air anomalies normally show a positive correlation with height are extremely sensitive to changes in height and gradients are large and unrelated to changes in underlying geological structure. Accordingly it is normal practice to smooth onshore free-air anomalies prior to contouring, the cut-off value o f the smoothing filter being usually determined by the 'wavelength' of the topography. A commonly used smoothing filter is the sine function (Bracewell 1965) given by the equation

sin n x ^ Ç

s m c x = --- 3.5

A perfect smoothing filter should not introduce errors, although, as in any averaging process, some o f the initial information is lost. If the distribution o f the observation points is irregular, aliasing errors may be introduced (St. John 1967). Because o f the difficulties associated with interpreting free-air anomalies onshore, they have not been used in this thesis. The free-air correction has been applied simply as an intermediate stage in the reduction to Bouguer anomaly.

3.1.3 The Bouguer correction

The Bouguer correction was developed by Bouguer (1749) as a simple

correction for the effect of the rock masses lying between the point o f observation and the reference surface. As noted above, these masses are the primary cause of the very high gradients observed in unsmoothed free-air anomalies. For a wide range of

topography the gravitational effect of the Bouguer model, an infinite horizontal slab of uniform density, is very close to reality, as can be demonstrated by some simple

calculations.

Firstly it can be shown that the gravitational effect Ag o f a cylinder ring element at any axial point (Fig. 3.3), distance h and H (h<H) from the plane limiting ring, is given by

Ag = 2!rpG [ ] 3.9

where G = 6.673x10-11 m^kg''s'^ is the gravitational constant, /othe density o f the ring

and r and R the inner and outer radii respectively. This equation is valid whether h and 77 are measured in the same direction or not, since only even powered terms are

involved. The physical reality underlying this mathematical result is that the effect at the origin of the element from -h to +h is always zero. With the notation shown in Fig. 3.3, the above equation can be rewritten as

Ag = 2npG ( k + l - m - n ) 3.10

For the Bouguer Plate both r and R are zero, and the difference between (if+ R^) and

R approaches zero as R approaches infinity, giving the simple Bouguer plate formula

g B = 27rpGH 3.11

An idea of the effect o f deviations o f the terrain from the Bouguer Plate can be obtained by comparing this expression with the field at the centre o f the upper surface o f a finite

disc. Table 3.2 shows the relationship between the ratio between R/Hand g / g B .

Table 3.2 Relationship between R/H and g / g B

R/H _g/gB

50 0.99

10 0.95

5 0.90

1 0.58

Since gB is itself directly proportional to //, the difference in milligals between g and gB increases rapidly with increase in the height of the block, for constant radius.

The model used above is, for certain, far from realistic, and in particular makes it easy to forget that topographic relief is as significant above the station as below it. For coastal stations, it is almost always topography above the level o f the station which is important. Since excess masses above station level and mass deficits below both result in an attenuation of field at the station, corrections for deviation o f the topography from the Bouguer Plate are always positive and never cancel. It is quite useful to compare the field due to a right circular cone at its own apex (Fig. 3.4) with gB, remembering the results can be applied to a station at the top o f a conical hill or at the bottom o f a conical depression.

Referring to Fig. 3.4 the gravity field at the apex due to the laminae of thickness dh may be obtained using the following equations

dg = iTtpG ((h + d h ) - h - l + n ) 3.12

now 1-n = dh siri(p thus

dg = 2npG (1 - sin<j)) dh integrating both sides to obtain

g = 2npG ( 1 -sin<j)) 3.13

The following table shows the relationship between ^ and g / g B

Table 3.3 Relationship between andg/g^ g^gB

35' 0.99

2°50' 0.95

5°25' 0.90

25° 0.58

Numerically the difference is proportional to h.

In regions o f low relief the simple Bouguer correction provides a satisfactory approximation to the effects of elevation and topography, provided only that an appropriate density is used. If this is not done the resulting Bouguer anomalies will remain markedly dependent on elevation, Nettleton's method o f determining the bulk density of near surface rocks (e.g. Nettleton 1939) being based on this fact. Simple Bouguer anomaly maps based on unsuitable value o f density can be very misleading if elevations vary widely and large changes occur in lateral density, but do serve important purpose. Since the simple Bouguer anomaly is derived solely from an assumed and constant density and from the four principal facts (longitude, latitude, height and observed gravity) o f the gravity station, the original data can in principle be recovered from the anomaly map. This will not be the case if further assumptions and corrections

are introduced. Useful re-interpretations have often been made on the basis of simple Bouguer anomaly maps when the original data have been lost or destroyed and the production and wide circulation of such maps still represents an important safeguard. The fact that all gravity stations in the Sorong Fault Zone Project were read within a few metres o f sea level virtually eliminates the effect o f density contrasts in the topography as a source o f error in interpretation.