3. Mass models of the Earth’s topography
3.3 Isostatic models
3.3.1 Airy-Heiskanen model
In 1855 the English astronomer G.B. Airy published his hypothesis of the compensation mechanism of topographic masses. He assumed that the Earth’s crust in regions with substantial topographic masses such as mountain chains in the scale of the Himalayas must sink with individual depth into the mantle. The weight of the topographic masses must be compensated by buoyant forces from below. According to his hypothesis, these forces result from the immersion of the lighter crust material into the denser mantle. Airy came already to the conclusion that the compensation can not be purely local and a certain part of the topography is carried by the resistance of the crust. This idea was enhanced later by Vening Meinesz in his regional compensation model. The model of Airy was mathematically described in 1924 and 1938 by W.A.
Heiskanen and after that used for the computation of different isostatic corrections. From that time the isostatic model was called the Airy-Heiskanen Model. It can be applied under the following assumptions:
1. the isostatic compensation takes place completely and locally, i.e. the compensation mass is directly under the regarded topographic mass and there is no regional emanating effect,
2. the density of the crust is assumed to be 2670kg/m3. The difference in the density between the crust and mantle is taken to be 600kg/m3.
The choice of the depth of the crust for the entire Earth,T, is extremely difficult (Fig. 3.2, left), since the crust thickness within the continents varies strongly; Heiskanen suggested a mean value ofT=30km. The mathematical treatment of this model can be performed by dividing the topographic masses into individual compartments, for which the hydrostatic balance condition is applied. This means that the pressure exerted by the weight of the topographic masses must be equal to the pressure produced by the lift forces of the isostatic masses. Also, the mass equality condition can be used for estimating the roots with thickness t under mountains or anti-roots with thickness t0 under oceans. In the following, the spherical and planar approximation will be studied.
Spherical approximation In the following, the geoid used as reference surface for the heights given by the DTMs is approximated by a geocentric mean sphere of radius R =6371km (Fig. 3.1). The geocentric radii of the computation and the integration points are given by adding the DTM heights, interpreted here as orthometric heights of these points, to the radius of the geocentric sphere, so that the ellipsoidal shape of the Earth is neglected. The error of this simplification is very small and in the size of approximately one percent of the total effect (cf. Novák and Grafarend 2005). In spherical approximation, a mass element of the topographic or compensation masses is calledtesseroid. The ideal situation of the Earth can be defined as follow: belowRthe density equals the density of the crustρcr, aboveRthe density equals0.0and below (R−T)the density equals the density of the mantleρm. The mass of the topography in this case is defined as the sum of all mass deviations (with respect to the ideal situation). The mass of a layered topography can
computation point computation point
Earth surface S Earth surface S
geoid geoid
P0
P
HP
HP
P
R P0
ellipsoid
Fig. 3.1: Approximation of the geoid by an ellipsoid (left) and a sphere (right)
be defined as the sum of all mass deviations with respect toρcr. Thus, assuming that a column consists of several layers with constant densities, the differences of the masses of these layers and the masses withρcr
give the topographic mass of this column (Fig. 3.2, left). The mass of the topography column can be given by (Classens 2003):
MST = Xi=l i=0
Z Z
σ R+HZ i
R
(ρi−ρi+1)ξ2dξdσ, (3.1)
with ρ0=
(0 forH0≥0
ρcr forH0≤0 and ρl+1=
(0 forHl>0
ρcr forHl<0. (3.2)
Also, the mass of the root with thicknesstor of the anti-root with thicknesst0 is given in spherical approxi-mation by
MSC= Z Z
σ RZ−T
R−T−t
∆ρr2drdσ, (3.3)
where∆ρis the difference between the density of the mantleρmand the density of the crust ρcr.
The condition of the “equilibrium of masses” between an element of mass at the surface of the Earth and a mass element at a depthT reads in spherical approximation
MST =MSC, (3.4)
or Xi=l i=0
Z Z
σ R+HZ i
R
(ρi−ρi+1)ξ2dξdσ= Z Z
σ RZ−T
R−T−t
∆ρξ2dξdσ. (3.5)
Fig. 3.2: Airy-Heiskanen model, left with constant density; several layers right
If the area elementsdσare assumed to be equal, Eq. (3.5) can be integrated with respected torto give the following equation:
Xl i=0
(ρi−ρi+1) 3
(R+Hi)3−R3
= ∆ρ 3
(R−T)3−(R−T−t)3
. (3.6)
This equation can be solved for the root columnst to give
t= (R−T)
1−
3
vu uu t1−
Pl i=0
[(R+Hi)3−R3] (ρi−ρi+1) (R−T)3∆ρ
. (3.7)
If the density of the topographic masses is assumed to be constant and identical to the density of the crust, Eq. (3.7) can be simplified for the roots as follows,
t= (R−T)
"
1− 3 s
1−[(R+H)3−R3]ρcr
(R−T)3∆ρ
#
, (3.8)
and for the anti-roots t0= (R−T)
"
3
s
1 + [R3−(R−H0)3] (R−T)3
(ρcr−ρw)
∆ρ −1
#
, (3.9)
whereρw is the density of the water.
Eq. (3.8) can be rearranged to give
t= (R−T)
1− 3 vu ut1−η
"
1 +H R
3
−1
#
, (3.10)
with
η:= R3 (R−T)3
ρcr
∆ρ (3.11)
Expanding the term inside the root of Eq. (3.10) into a binomial series, η
"
1 +H R
3
−1
#
=η 3H
R +3H2 R2 +H3
R3 . . .
=:ζ, (3.12)
Eq. (3.10) can be written as t= (R−T)h
1−(1−ζ)1/3i
. (3.13)
Also, expanding the term(1−ζ)1/3in Eq. (3.13) in a binomial series and restricting the series to the quadratic term gives after some rearrangements
t= R
R−T 2
ρcr
∆ρH
"
1 +H R
R R−T
3
ρcr
∆ρ+ 1
!#
+O H
R 3
. (3.14)
Applying an analogous procedure for the anti-roots, Eq. (3.9) can be approximated as t0=
R R−T
2
(ρcr−ρw)
∆ρ H0
"
1 + H0 R
R R−T
3
(ρcr−ρw)
∆ρ −1
!#
+O H0
R 3
. (3.15)
The roots and anti-roots in spherical approximation with constant topographic densities have been derived also by Heiskanen and Vening Meinesz (1958), Rummel et al. (1988) and Abd-Elmotaal (1991).
A frequently used concept to reduce the several topographic layers to only one layer for reducing CPU time is the concept of “equivalent rock topography”. In this approach, all layers consisting of material with different densities are replaced by an equivalent rock layer with the density of the crust ρcr and a certain thickness.
The thickness of the equivalent layerHeqis determined by assuming the condition that the mass of a column of equivalent rock layer should be equal to the mass of the original layers in the same column so that it holds (see Fig. 3.3):
MST = Xl i=0
(ρi−ρi+1) 3R2
(R+Hi)3−R3
dσ= ρcr
3R2
(R+Heq)3−R3
dσ. (3.16)
Eq. (3.16) can be solved for the quantityHeq (rock equivalent topography):
Fig. 3.3: Ideal situation of the topography (left), three layers of topography (middle) and rock equivalent topography (right)
Heq =
R3+3R2 ρcr
MST 13
−R. (3.17)
If the equivalent rock topography is used, the roots can be given in spherical approximation by inserting the rock equivalent topography in Eq. (3.14):
teq= R
R−T 2
ρcr
∆ρHeq
"
1 + Heq
R
R R−T
3
ρcr
∆ρ+ 1
!#
+O Heq
R 3
. (3.18)
Planar approximation In case of the planar approximation, the mass of the topography and the com-pensating masses are given by (Fig. 3.4),
MPT = Xl i=0
(ρl−ρl+1)Hidσ (3.19)
and
MPC= ∆ρtdσ, (3.20)
respectively. Therefore, the root or anti-root is given by t= 1
∆ρ Xl i=0
(ρl−ρl+1)Hi. (3.21)
The formulae for the roots and anti-roots in planar approximation and by assuming constant density can
ρ ρ ρ
ρ
ρ
Fig. 3.4: Airy-Heiskanen isostatic model in planar approximation be given as the limits of the functions in Eq. (3.14) and Eq. (3.15) withR→ ∞:
t= lim
R→∞
R R−T
2
ρcr
∆ρH
"
1 +H R
R R−T
3
ρcr
∆ρ+ 1
!#
+ O H
R 3!
= ρcr
∆ρH (3.22)
t0 = lim
R→∞
R R−T
2
(ρcr−ρw)
∆ρ H0
"
1 +H0 R
R R−T
3
(ρcr−ρw)
∆ρ −1
!#!
=(ρcr−ρw)
∆ρ H0.
(3.23)
Also the rock equivalent topography in planar approximation is given by:
MPT = Xl i=0
(ρi−ρi+1)Hidσ=ρcrHeqdσ, (3.24)
which follows (assuming also the areas of the elements to be equal):
Heq = Pl i=0
(ρi−ρi+1)Hi
ρcr
. (3.25)
If the topography has a constant density ρcr (one layer topography), the rock equivalent topography can be derived by
Heq =
land:Heq=H
ocean:Heq= ρcr−ρw
ρcr
H0
. (3.26)
In this case the roots or anti-roots are given by teq= ρcr
∆ρHeq. (3.27)