Instruments for gravity measurement fall into two types: Those forabsolute gravity measurement and those for relative gravitymeasurement.

Figure 1 shows a number representing g with the numbers ofsignificant figures required for various effects to be determined.

Figure: Schematic of the relative magnitudes and precision required formeasurements.

Figure: Historical development of the relative errors achievable in gravitymeasurements since 1600. After: Torge, 1989.

Figure 2 shows the steady improvement of precision of gravitymeasurements since 1600. The kink in the slope of improvementcorresponds to the newer technologies made possible by the use ofelectronics and digital computers.

Figure: A pendulum of length l and maximum deflection α0.

The period of a pendulum is given by:

Tideal = 2π

(

l

g

)1

2

(

1 +α2

0

16

)

(1)

where Tideal is the period, l is the length, g the acceleration dueto gravity and α0 is the angle of the swing, see figure 3.

State of the art absolute gravity measurements can be made with afree-fall gravimeter; see figure 4.

Figure: Diagram of the principle of a free fall gravimeter.

A falling mass passes through successive points z0, z1, . . . atsuccessive times t0, t1, . . ..From Newton’s second law of motion, we have:

mg = md2z

dt2(2)

which can be integrated to give:

z = z0 + v0t +1

2gt2 (3)

where z0 is the starting point, v0 is the initial velocity and t is time.

We can write, for three times t1, t2 and t3:

z1 = z0 + v0t1 +1

2gt2

1 (4)

z2 = z0 + v0t2 +1

2gt2

2

z3 = z0 + v0t3 +1

2gt2

3

from which:

g = 2(z3 − z1)(t2 − t1) − (z2 − z1)(t3 − t1)

(t3 − t1)(t2 − t1)(t3 − t2)(5)

Modern instruments use up to 200 readings of t and z .

Example of accuracy:∆z = 50cm and ∆t = 0.3 s to get ∆g/g = 10−9 implying that∆z must be measured to 0.5 × 10−9m and ∆t to 0.2 nsecond.This assumes that g is constant — but it is not. We know thegradient is −0.3086 mGal/m. We could incur an error of 0.15mGal.We must also be careful of magnetic fields, microseismicity etc.The cost is of the order of tens of thousands of pounds.The FG5 instrument is capable of 2 parts in 1 billion (109) or 0.2µGals.The rise and fall instrument FG5L has advantages: symmetry of airdrag and magnetic field drag on upward and downward legs. Notime loss in raising mass.

Figure: Diagram of the principle of a gravimeter in which the force on amass is measured by the extension of a spring.

Balance g with another force. E.g. spring, superconductinggravimeter, which uses super conducting coils to keep a niobiumsphere in place, see figure 8 for the principle.

Second coils apply current to compensate for g .

mg = k(l − l0)

implies g =k

m(l − l0) (6)

where k is the spring constant, which is a function of shape andelasticity.This method is not directly applicable, it is hard to find k, l0 etc.

For two different positions A and B :

at Ag1 =k

m(l1 − l0)

at Bg2 =k

m(l2 − l0)

giving:

∆g = g2 − g1 =k

m∆l (7)

The accuracy is 10−8 to 10−9g i.e. 1 to 10 µGal. We also need toknow ∆h, the difference in altitude between A and B .Temperature changes l0 and we also need k/m.The Scintrex meter uses this principle (i.e. vertical spring) but usesan electrostatic force (feedback) to keep the mass in the sameplace. The advantage is that the tension in the spring remainsconstant.Relative gravity can also be used to measure the vertical gradientto improve the absolute measurements.

The acceleration due to gravity is controlled by:

1. Geological controls

1.1 Density of subsurface mass.1.2 Distance from measurement point.1.3 Volume of mass.

2. General controls

2.1 Latitude2.2 Height2.3 Topography around measurement site.2.4 Time - e.g. lunar.

If we did not correct for the general controls it would not bepossible to use gravity to investigate the geological controls sincethe effects of the general controls are of much greater magnitudethan those of geological controls.There are different ways of accounting for the general controls andresulting variations are called:

1. Free air anomaly FAA

2. Bouguer anomaly BA

3. Isostatic anomaly IA

FAA = gobs − gth + Free air correction (FAC) (8)

where gobs is the vertical component of g , gth is the theoretical or“normal” value of g at mean sea level at that latitude. Removesmajor component of g leaving only local effects. The FAC correctsfor the height above the mean sea level. (0.3086 mGals per metre)

δg

g= −2

δr

r(9)

We can take height datum as mean sea level, or any other datum,e.g. lowest point in a survey.

FAA = gobs − gth + 0.3086h (10)

with the result in mGal.

Figure (supplied separately) shows FAA and gobs. It does not takethe rock mass into account so the FAA is not normally used forland-based studies. Main use is at sea.

BA = FAA − Bouguer correction (BC) (11)

The Bouguer anomaly corrects for the rock mass between themeasuring site and the height datum.The correction assumes rock to be a flat infinite slab of thickness h

and constant density ρ.

BC = 2πGρh (12)

Note that BC is always positive. This is why it is subtracted in theBA formula.

NB This is an approximation!. It is called the simple Bougueranomaly (SBA).When a terrain correction (TC) is applied one obtains theComplete Bouguer Anomaly (CBA).

CBA = SBA + TC (13)

This accounts for the fact that the rock below the observationpoint is not an infinite slab.

Figure: Airy Heiskanen isostasy.

t =hρ

∆ρ+ T (14)

where t is the depth of the deflection below sea level, h is theelevation, ρ is the crustal density, ∆ρ is the Moho density contrastand T is the crustal thickness at sea level.The deflection is the elevation times the amplification factor.

Sea levelTopography

Airy-Heiskanen

root

(compensation depth)

h

d

T crust

mantle

ρ

ρ + ∆ρ

Figure: Airy Heiskanen isostasy, with deflection on Moho defined.

d(ρ + ∆ρ) + Tρ = (T + d + h)ρ

d∆ρ = hρ

d = hρ

∆ρ

E.g. for the Andes ρ = 2.67 gm/cm3 and ∆ρ is highly variable butis in the range 0.2 − 0.6 gm/cm3.USGS program AIRYROOT will calculate the effect of Airycompenstated MOHO. It is best to calculate out to 400km.NB1 Amplification >> 1NB2 Assume no internal strength to crust, each block isindependent of the adjacent one.

Figure: Pratt isostasy, of historical interest only.

Topography is compensated by crustal ρ variations. Of historicalinterest in the absence of other geological information.

Figure: Strong crust or lithosphere

This is an extreme end-member.

Figure: Flexure of Pacific plate lithosphere under loading from Hawaii

This takes into account the finite strength of crust/lithosphere.