Gravity - The Solid Earth - page 202

from the compositional layers, crust and mantle, which have been discussed
previously.
The depth below which all pressures are hydrostatic is termed the compensa-
tion depth. At or below the compensation depth, the weight of (imaginary) verti-
cal columns with the same cross-sectional area must be the same. A mountain in
isostatic equilibrium is therefore compensated byamass deficiency between the
surface and the compensation depth. In contrast, an ocean basin in isostatic equi-
librium is compensated by extra mass between the seabed and the compensation
depth.

Airy's hypothesis
In this hypothesis the rigid upper layer and the substratum are assumed to have
constant densities, ρ s , respectively. Isostatic compensation is achieved by

mountains having deep roots (exactly like an iceberg). Figure 5.6(a) illustrates
this hypothesis. Taking an arbitrary compensation depth that is deeper than the
deepest mountain root in the substratum and equating the masses above that depth
in each vertical column of unit cross-sectional area, one obtains

ρ u and

t ρ u + r 1 ρ s = ( h 1 + t + r 1 ) ρ u
= ( h 2 + t + r 2 ) ρ u + ( r 1 − r 2 ) ρ s
= d ρ ω + ( t − d − r 3 ) ρ u + ( r 1 + r 3 ) ρ s (5.23)

A mountain of height h 1 would therefore have a root r 1 given by

h 1 ρ u
ρ s − ρ u

r 1 = (5.24)

Similarly, a feature at a depth d beneath sea level would have an anti-root r 3 given
by

d ( ρ u − ρ w )

ρ s − ρ u

r 3 = (5.25)

The rigid upper layer (lithosphere) has density ρ u ,but Eqs. (5.23)-(5.25)

apply equally well when ρ u is replaced by ρ c (the density of the crust) and

ρ s is replaced by ρ m (the density of the mantle). This is because the crust-

mantle boundary is embedded in and is part of the lithosphere, so loading at the
surface and subsequent deflection of the base of the lithosphere deflects the crust-
mantle boundary. Furthermore, the difference between the density of the mantle
at the crust-mantle boundary and the density of the mantle at the lithosphere-
asthenosphere boundary may be very small (see Section 8.1.2). Therefore, Eqs.
(5.24) and (5.25) are often applied in the following forms:

h 1 ρ c
ρ m − ρ c

r 1 =

(5.26)

and

d ( ρ c − ρ w )
ρ m − ρ c

r 3 = (5.27)