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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):
dζ
=
∂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
dζ
=
−
εds
−
dh .
(8-149)
γ
earth's surface
dh
W=
P
P
@³
@
s
ds
"
ds
U=
P
Fig. 8.16. Astronomical leveling according to Molodensky