Geoscience Reference
In-Depth Information
H
∗
=
C
0
dC
γ
,
(4-54)
C
=
γH
∗
,
(4-55)
where
H
∗
1
H
∗
γdH
∗
γ
=
(4-56)
0
is the mean normal gravity along the plumb line.
As the normal potential
U
is a simple analytic function, these formulas
can be evaluated straightforwards; but since the potential of the earth is
evidently not normal, what does all this mean? Consider a point
P
on the
physical surface of the earth. It has a certain potential
W
P
and also a certain
normal potential
U
P
, but in general
W
P
=
U
P
. However, there is a certain
point
Q
on the plumb line of
P
, such that
U
Q
=
W
P
;thatis,thenormal
potential
U
at
Q
is equal to the actual potential
W
at
P
. The normal height
H
∗
of
P
is nothing but the ellipsoidal height of
Q
above the ellipsoid, just
as the orthometric height of
P
is the height of
P
above the geoid.
For more details the reader is referred to Sect. 8.3; Fig. 8.2 illustrates
the geometric relations.
We now give some practical formulas for the computation of normal
heights from geopotential numbers. Writing (4-56) in the form
H
∗
1
H
∗
γ
=
γ
(
z
)
dz
(4-57)
0
corresponding to (4-28), then we can express
γ
(
z
) by (2-215) as
γ
(
z
)=
γ
1
a
2
z
2
,
a
1+
f
+
m
2
f
sin
2
ϕ
z
+
2
3
−
−
(4-58)
where
γ
is the gravity at the ellipsoid, depending on the latitude
ϕ
but not
on
z
. Thus, straightforward integration with respect to
z
yields
γ
z
1+
f
+
m
2
f
sin
2
ϕ
z
2
2
z
3
3
1
H
∗
2
a
3
a
2
H
∗
γ
=
−
−
+
0
(4-59)
γ
H
∗
−
a
2
H
∗
3
a
1+
f
+
m
2
f
sin
2
ϕ
H
∗
2
+
1
H
∗
1
1
=
−
or
γ
=
γ
1
a
2
.
1+
f
+
m
2
f
sin
2
ϕ
H
∗
2
a
+
H
∗
2
−
−
(4-60)
This formula may be used for computing
H
∗
by the formula
H
∗
=
C
γ
.
(4-61)
