Global Positioning System Reference
In-Depth Information
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TABLE 9.3
Relative Distance Error versus Height
h
m[m]
h
m
/R
6.37
1:1000000
63.7
1:100000
100
1:64000
500
1:13000
637
1:10000
1000
1:6300
th
at is measured together with the slant distance. For observations reduced in such a
manner,
∆
h
is small (although not zero), but there is now a corresponding accuracy
re
quirement regarding the measured elevation angle.
If both stations are located at about the same height
h
1
≈
h
2
≈
h
m
, one obtains
[32
from (e)
h
m
R
s
− µ
µ
≈
(9.2)
Lin
—
0.3
——
Nor
PgE
This equation relates the relative error in distance reduction to the mean height of
the line. Table 9.3 shows that just6minheight error causes a 1 ppm error in the
reduction. Networks are routinely achieved that accurately with GPS.
Since modern EDM instruments are very accurate, it is desirable to apply the
height corrections consistently. It is good to remember the rule of thumb that
a6m
error in height of the line causes a relative change in distance of 1 ppm.
We recognize
that geodetic heights are required, not orthometric heights. Since geoid undulations
can be as large as 100 m, it is clear that they must be taken into account for high-
precision surveying.
[32
9.
1.2 Direct and Inverse Solutions on the Ellipsoid
The reductions discussed above produce the geodesic observables, i.e., the geodesic
azimuths
δ
, the geodesic distance
s
, and the angle between geodesics
. At the heart of
computations on the ellipsoidal surface are the so-called direct and inverse problems,
which are summarized in Table 9.4. For the direct problem, the geodetic latitude and
longitude of one station, say,
P
1
(ϕ
1
,
α
λ
1
)
, and the geodesic azimuth
α
12
and geodesic
distance
s
12
to another point
P
2
are given; the geodetic latitude and longitude of
station
P
2
(ϕ
2
,
λ
2
)
, and the back azimuth
α
21
must be computed. For the inverse
problem, the geodetic latitudes and longitudes of
P
1
(ϕ
1
,
λ
2
)
are given,
and the forward and back azimuth and the length of the geodesic are required. Note
that
λ
1
)
and
P
2
(ϕ
2
,
s
12
s
21
but
α
12
α
21
±
180°. There are many solutions available in the
literature for the direct and inverse problems. Some of these solutions are valid for
geodesics that go all around the ellipsoid. We use the Gauss midlatitude (GML)
functions given in Table B.2.
Because the GML functions are a result of series developments and, as such,
subject to truncation errors, Figure 9.5 has been prepared to provide some insight into
=
=
