Geoscience Reference
In-Depth Information
The normal height
H
∗
, and hence the telluroid Σ, can be determined by
leveling combined with gravity measurements, according to Sect. 4.4. First
the geopotential number of
P
,
C
=
W
0
−
W
P
, is computed by
C
=
P
0
gdn,
(8-20)
where
g
is the measured gravity and
dn
is the leveling increment. The normal
height
H
∗
is then related to
C
by an
analytical
expression such as (4-63),
1+(1+
f
+
m
2
,
+
C
aγ
Q
0
C
γ
Q
0
C
aγ
Q
0
H
∗
=
2
f
sin
2
ϕ
)
−
(8-21)
where
γ
Q
0
is the normal gravity at the ellipsoidal point
Q
0
.Notethat
H
∗
is
independent of the density.
The normal height
H
∗
of a ground point
P
is identical with the ellipsoidal
height
h
, the height above the ellipsoid, of the corresponding telluroid point
Q
. If the geopotential function
W
were equal to the normal potential function
U
at every point, then
Q
would coincide with
P
, the telluroid would coincide
with the physical surface of the earth, and the normal height of every point
would be equal to its ellipsoidal height. Actually, however,
W
P
=
U
P
; hence
the difference
=
h
P
− H
P
ζ
P
=
h
P
− h
Q
(8-22)
is not zero. This explains the term “height anomaly” for
ζ
.
The gravity anomaly is now defined as
∆
g
=
g
P
−
γ
Q
;
(8-23)
it is the difference between the actual gravity as measured on the ground
and the normal gravity on the telluroid. The normal gravity on the telluroid,
which we shall briefly denote by
γ
, is computed from the normal gravity at
the ellipsoid,
γ
Q
0
, by the normal free-air reduction, but now applied
upward
:
γ ≡ γ
Q
=
γ
Q
0
+
∂γ
∂
2
γ
1
2!
∂h
H
∗
+
∂h
2
H
∗
2
+
··· .
(8-24)
For this reason, the new gravity anomalies (8-23) are called
free-air anoma-
lies
.Theyare
referred to ground level
, whereas the conventional gravity
anomalies have been referred to sea level. Therefore, the new free-air anoma-
lies have nothing in common with a free-air reduction of actual gravity to
sea level, except the name. This distinction should be carefully kept in mind.
A direct formula for computing
γ
at
Q
is (2-215),
1
2
,
+3
H
∗
a
2
f
sin
2
ϕ
)
H
∗
a
γ
=
γ
Q
0
−
2(1 +
f
+
m
−
(8-25)
