Geoscience Reference
In-Depth Information
good and bad points of each method. Polynomial fitting will be used to explain the
basics of GPS leveling because it is readily understandable.
If the quadratic polynomial is used as the fitting model of the height anomaly,
then the height anomaly of this region
ʶ
can be expressed by:
L
2
B
2
,
ʶ ¼ ʱ
0
þ ʱ
1
ʔ
L
þ ʱ
2
ʔ
B
þ ʱ
3
ʔ
þ ʱ
4
ʔ
L
ʔ
B
þ ʱ
5
ʔ
ð
4
:
44
Þ
where
B
0
are the differences between the geodetic
longitude and latitude L, B of the point to be calculated and L
0
, B
0
of a known point
in this region;
ʔ
L
¼
L
L
0
and
ʔ
B
¼
B
5) is the undetermined coefficient. Suppose the number
of GPS-leveling points in this region is n, meaning that
ʱ
i
(i
¼
0
˂
i
(i
¼
1
n, n
6) is
already known; then n equations can be formed by (
4.44
). Let:
0
@
1
A
0
@
1
A
0
@
1
A
L
1
B
1
ʱ
0
ʱ
1
ʱ
2
ʱ
3
ʱ
4
ʱ
5
1
ʔ
L
1
ʔ
B
1
ʔ
ʔ
L
1
ʔ
B
1
ʔ
ʶ
1
ʶ
2
ʶ
3
ʶ
4
B
2
1 ʔL
3
ʔB
3
ʔL
3
ʔL
3
ʔB
3
ʔB
3
1 ʔL
4
ʔB
4
ʔL
4
ʔL
4
ʔB
4
ʔB
4
⋮⋮⋮⋮ ⋮ ⋮
1
L
2
1
ʔ
L
2
ʔ
B
2
ʔ
ʔ
L
2
ʔ
B
2
ʔ
!
!
ʱ
¼
X
¼
ʶ
¼
:
⋮
L
n
B
n
ʔ
L
n
ʔ
B
n
ʔ
ʔ
L
n
ʔ
B
n
ʔ
ʶ
n
ð
4
:
45
Þ
!
is:
Then the least square solution of the undetermined coefficient vector
ʱ
1
X
T
P
!
,
!
X
T
PX
ʱ
¼
ʶ
ð
4
:
46
Þ
where P is the weight matrix. If the known data are considered as mutually
independent,
then the principal diagonal element of P is the weight of
˂
i
(i
n).
After the determination of the undetermined coefficient, the value of
¼
1
ʶ
of the
unknown point can be calculated according to L, B of this point by (
4.44
).
This method can also be used to determine the geoid height N by replacing H
N
with the orthometric height H
O
of the GPS-leveling point and
with N in (
4.43
).
At present, centimeter-level or even higher-level precision can be achieved in
determining the height anomaly by the method of GPS leveling combined with
gravity data.
From the above, it is known that GPS leveling can determine the geoid directly
on land. On the ocean, the geoid can be deduced directly by satellite altimetry (see
Chap.
2
). The shape of the geoid can be deduced directly now, whereas it could only
be obtained by gravity field in the period of classical geodetic survey.
The separation of geoid from ellipsoid N (geoid height) and the height anomaly
ʶ
ʶ
can also be determined based on the Earth gravity field model. Put more directly,
the location of the geoid is related to the gravity potential, whereas the location of
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