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upward from
Q
0
to
Q
. Whereas for the first process the use of the normal
gradient
∂γ/∂h
is problematic, it is fully justified for the second process.
In a similar way, we might interpret
δϕ
normal
as a reduction of
ϕ
for
normal curvature of the plumb line upwards, say, from
P
0
to
P
.Thisis
possible because in (8-136)
ϕ
could be said to refer to
P
0
(because
P
0
and
P
0
practically coincide), and because
ϕ
denotes the latitude of the tangent
to the normal plumb line at
P
. This interpretation is instructive because of
the analogy with gravity reduction, though regarding
ϕ
and
ϕ
as ellipsoidal
and dynamic latitude of the same point
P
appears more natural. Refer again
to our key figure (Fig. 8.13).
As pointed out above, the present interpretation of
ξ
c
and
η
c
as isostati-
cally reduced deflections of the vertical at the earth's surface is conceptually
rigorous and therefore also practically more accurate, but this decisive ad-
vantage implies a computational drawback if integration along a profile is
used: Since this integration must now be performed along the earth's surface
and not along a level surface such as the geoid, computation will be more
complicated. Instead of the simple Helmert formula (8-143), we now must
use the Molodensky formula (8-150):
B
B
g
c
− γ
γ
ζ
c
ζ
c
ε
c
ds
B
−
A
=
−
−
dh
(8-158)
A
A
with
ε
c
=
ξ
c
cos
α
+
η
c
sin
α,
(8-159)
and ∆
g
c
=
g
c
γ
,where
g
c
is the isostatically reduced surface value of gravity
(measured value
g
minus attraction of the topographic-isostatic masses).
From the isostatic height anomalies
ξ
c
obtained in this way, we then get
the actual height anomalies
ζ
by applying the indirect effect:
−
ζ
=
ζ
c
+
δζ
(8-160)
with
δζ
=
T
TI
γ
.
(8-161)
This is completely analogous to (8-5) and (8-3), but now
T
TI
is the potential
of the topographic-isostatic masses at the surface point
P
. As a matter of
fact, normal gravity in (8-3) refers to the ellipsoid, and in (8-161) to the
telluroid, but the difference is generally small.
For higher mountains, the isostatic reduction procedure described in the
present section is preferable in practice to a direct application of Moloden-
sky's formula (8-150) because the isostatically reduced vertical deflections
