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or, approximately,
ε 0 A + ε 0 B
2
N B
N A =
s AB ,
(8-144)
where s AB denotes the horizontal distance between A and B .Theminus
sign is conventional. Cf. Sect. 5.14.
A corresponding relation to height anomalies according to Molodensky
is found as follows (Molodensky et al. 1962: p. 125):
= ∂ζ
∂s ds + ∂ζ
∂h dh ,
(8-145)
notations following Fig. 8.16. Since the earth's surface is not a level surface,
we also have a vertical part ( ∂ζ/∂h ) h in addition to the usual horizontal
part ( ∂ζ/∂s ) ds . The vertical part arises from change in height and is usually
smaller than the horizontal part.
In analogy to (8-142), the horizontal part is given by
∂ζ
∂s =
ε,
(8-146)
where ε denotes the dynamical deflection of the vertical at the earth's surface;
cf. (8-140) and Fig. 8.13. For the vertical part we have from (8-126):
T
γ
=
∂T
∂h
∂h T
∂ζ
∂h =
∂h
1
γ
1
γ
∂γ
(8-147)
or
∂ζ
∂h =
g
γ
g
γ
=
(8-148)
γ
according to the fundamental equation of physical geodesy (8-36).
Hence (8-145) becomes
g
γ
=
εds
dh .
(8-149)
γ
earth's surface
dh
W= P
P
@ s ds
"
ds
U= P
Fig. 8.16. Astronomical leveling according to Molodensky
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