## Tests of isostasy

The state of isostatic compensation of a region can be assessed by making use of gravity anomalies. The isostatic anomaly, IA, is defined as the Bouguer anomaly minus the gravity anomaly of the subsurface compensation. Consider a broad, flat plateau of elevation h compensated by a root of thickness r. The terrain correction

From the Airy criterion for isostatic equilibrium:

Substitution of this condition into the equation reveals that the isostatic anomaly is equal to the free-air anomaly over a broad flat feature, and this represents a simple method for assessing the state of isostatic equilibrium. Figure 2.33 shows free-air, Bouguer and isostatic anomalies over a broad flat feature with varying degrees of compensation. Although instructive in illustrating the similarity of free-air and isostatic anomalies, and the very different nature of the Bouguer anomaly, this simple Airy isostatic anomaly calculation is clearly unsatisfactory in not taking into account topography and regional compensation due to flexure of the lithosphere.

To test isostasy over topographic features of irregular form more accurate computation of isostatic anomalies is required. This procedure involves calculating the shape of the compensation required by a given hypothesis of isostasy, computing its gravity anomaly, and then subtracting this from the observed Bouguer anomaly to provide the isostatic anomaly. The technique of computing the gravity anomaly from a hypothetical model is known as forward modeling.

Gravity anomalies can thus be used to determine if a surface feature is isostatically compensated at depth. They cannot, however, reveal the form of compensa

Figure 2.33 Free air, Bouguer and Airy isostatic anomalies over an idealized mountain range (a) in perfect isostatic equilibrium, (b) with 70% isostatic compensation, (c) with 30% isostatic compensation, (d) uncompensated. Densities in Mg m-3.

tion and indicate which type of mechanism is in operation. This is because the compensation occurs at a relatively deep level and the differences in the anomalies produced by a root/antiroot (according to the Airy hypothesis) or by different density units (according to the Pratt hypothesis) would be very small. Moreover, the gravity anomalies over most regions contain short wavelength components resulting from localized, uncompensated geologic structures that obscure the differences in the regional field arising from the different forms of compensation.

A more sophisticated test of isostasy involves the spectral analysis of the topography and gravity anomalies of the region being studied (Watts, 2001). The relationship between gravity and topography changes with wavelength. Moreover, the way in which it changes varies for different isostatic models. Thus by determining the frequency content of the gravity and topographic data it is possible to determine the type of compensation pertaining in the area. The technique also yields an estimate of Te, the elastic thickness of the lithosphere (Sections 2.11.4, 2.12).

Figure 2.34 Bouguer and isostatic gravity anomalies and their relation to seismic velocity sections from the western USA. Velocities in km s- (redrawn from Garland, 1979).

Information on the geometric form of isostatic compensation can also be gained by a combined analysis of gravity and seismic refraction data, as the latter technique can provide a reasonably detailed picture of the sub-surface structure of the region under consideration. Such studies have demonstrated that the broad isostatic equilibrium of continents and oceans is mainly accomplished by variations in crustal thickness according to the Airy hypothesis. Figure 2.34 shows seismic velocity sections from the western USA in which surface topography is largely compensated by Moho topography, although in several locations density variations in the upper mantle must be invoked to explain the isostatic compensation. A cross-section of the western USA (Fig. 2.35) reveals, however, that crustal thickness is not necessarily related to topographic elevation as the Great Plains, which reach a mean height of 1 km, are underlain by crust 45-50 km thick and the Basin and Range Province, at an average of 1.2 km above mean sea level, is underlain by a crustal thickness averaging 25-30 km (Section 7.3). Clearly, the Basin and Range Province must be partially compensated by a Pratt-type mechanism resulting from the presence of low density material in the upper mantle. Similarly, ocean ridges (Section

6.2) owe their elevation to a region of low density material in the upper mantle rather than to a thickened crust.

There are regions of the Earth's surface that do not conform to the concepts of isostasy discussed here. The hypotheses discussed above all assume that the support of surface features is achieved by their attaining hydrostatic equilibrium with the substrate. In certain areas, however, in particular convergent plate margins, surface features are supported dynamically by horizontal stresses. Such features provide the largest isostatic anomalies observed on the Earth's surface.