Gravity - The Solid Earth - page 207

of ∼ 3 km. If the ridge were uncompensated, the gravity anomaly should increase

by

2700 − 1000
1000

42 × 3 × ≈ 200 mgal

Since the observed anomaly increases by only 20 mgal, the ridge must be

compensated. In contrast, Figs. 5.13 and 9.64 show the gravity anomaly across
Hawaii. An uncompensated decrease in water depth of

∼

∼ 4kmwould cause an

increase in gravity of

2700 − 1000
1000

42 × 4 × ≈ 300 mgal

This matches the observed gravity anomaly and so confirms that Hawaii is not
compensated. The excess mass must therefore be supported by the Pacific plate
upon which it rests (see Section 5.7.1).
The simplest way to determine whether a large-scale structure such as a moun-
tain chain or large sedimentary basin is in isostatic equilibrium is therefore to use
the free-air anomaly. If a structure or region is totally compensated, the free-air
anomaly is very small away from the edges of the structure, provided that the
structure is at least about ten times wider than the compensation depth. If the
structure is only partially compensated, or not compensated at all, then the free-
air anomaly is positive, perhaps up to several hundred milligals in magnitude,
depending on the structure and degree of compensation. For a totally or partially
compensated structure the Bouguer anomaly is negative, whereas for an uncom-
pensated structure the Bouguer anomaly is zero. Free-air anomalies are almost
isostatic anomalies. They do not imply any specific mechanism for compensation
but are small if compensation is complete.
Take, for example, the mountains illustrated in Fig. 5.6(a) and assume them to
be in isostatic equilibrium with the left-hand column of crust (of thickness t ). The
Bouguer anomaly across these mountains will be negative since below sea level
there is a mass deficiency beneath the mountains. The attraction of the excess
mass of the mountains that balanced this mass deficiency has been removed in
calculating this anomaly. In contrast, the free-air anomaly over the mountains
is positive and much smaller in magnitude. The free-air anomaly is positive
over the mountains rather than being zero, because the mountains are closer to
the measurement point than is their deep compensating structure. Even though
the mass deficit of the compensating structure is the same as the mass excess
of the mountains, its negative contribution to the gravity at the measurement
point is smaller because it is further from the measurement point.
Another way to determine whether a structure or region is in isostatic equi-
librium is to propose a series of density models and then calculate the Bouguer
anomaly that each would give. The isostatic anomaly for the region is then the
actual Bouguer anomaly minus the computed Bouguer anomaly for the proposed
density model. Thus, each density model for the region has a different isostatic
anomaly.