Geoscience Reference
In-Depth Information
The astronomical coordinates, latitude Φ and longitude Λ, form two of
the three spatial coordinates of
P
. As third coordinate we may take the
orthometric height
H
of
P
or its potential
W
. Equivalent to
W
is the
geopo-
tential number C
=
W
0
− W
,where
W
0
is the potential of the geoid. The
orthometric height
H
was defined in Sect. 2.2; see also Fig. 2.2. The relations
between
W
,
C
,and
H
are given by the equations
H
W
=
W
0
−
gdH
=
W
0
−
C,
0
W
=
H
0
C
=
W
0
−
gdH,
(2-56)
W
=
C
0
dW
g
dC
g
H
=
−
,
W
0
which follow from integrating (2-21). The integral is taken along the plumb
line of point
P
, starting from the geoid, where
H
=0and
W
=
W
0
(see also
Fig. 2.8).
The quantities
Φ
,
Λ
,W
or Φ
,
Λ
,H
(2-57)
are called
natural coordinates
. They are the real-earth counterparts of the
ellipsoidal coordinates. They are related in the following way to the geocen-
tric rectangular coordinates
x, y, z
ofSect.2.1.The
x
-axis is associated with
the mean Greenwich meridian; from Fig. 2.7 we read that the unit vector of
the vertical
n
has the
xyz
-components
n
= [cos Φ cos Λ
,
cos Φ sin Λ
,
sin Φ] ;
(2-58)
the gravity vector
g
is known to be
g
=[
W
x
,W
y
,W
z
]
.
(2-59)
earth's surface
level surface
P
W = constant
H
leve
l
geoid
W=
0
0
Fig. 2.8. The orthometric height
H
