Geoscience Reference
In-Depth Information
B
A
Fig. 4.3. Two different leveling lines connecting
A
and
B
(taken to-
gether, they form a circuit)
The symbol
denotes an integral over a circuit.
On the other hand, the measured height difference, that is, the sum of
the leveling increments
δn
=
B
A
∆
n
AB
=
B
dn ,
(4-8)
A
depends on the path of integration and is, thus, not in general zero for a
circuit:
dn
=misclosure
=0
.
(4-9)
In mathematical terms,
dn
is not a perfect differential (the differential of a
function of position), whereas
dW
=
−gdn
is perfect, so that
dn
becomes a
perfect differential when it is multiplied by the
integrating factor
(
g
).
Thus, potential differences are the result of leveling combined with grav-
ity measurements. They are basic to the whole theory of heights; even ortho-
metric heights must be considered as quantities derived from potential differ-
ences. Leveling without gravity measurements, although applied in practice,
is meaningless from a rigorous point of view, for the use of leveled heights
(4-8) as such leads to contradictions (misclosures); it will not be considered
here.
−
4.2
Geopotential numbers and dynamic heights
Let
O
be a point at sea level, that is, simplifying speaking, on the geoid;
usually a suitable point on the seashore is taken. Let
A
be another point,
connected to
O
by a leveling line. Then, by formula (4-6), the potential
difference between
A
and
O
can be determined. The integral
A
gdn
=
W
0
−
W
A
=
C,
(4-10)
0
