Global Positioning System Reference
In-Depth Information
Example 8.2
To emphasize the fundamental impact of the matrix
F
in (8.36) we
shall determine the elevation angle for a satellite. The local
topocentric system
uses three unit vectors
(
e
,
n
,
u
)
=
(
east
,
north
,
up
)
. Those are the columns of
F
.
The vector
r
between satellite
k
and receiver
i
is
=
X
k
Z
i
.
Y
k
Z
k
r
−
X
i
,
−
Y
i
,
−
The unit vector in this satellite direction is
ρ
=
r
/
r
. Then Figure 8.10 gives
T
e
ρ
=
sin
α
sin
z
,
T
n
ρ
=
cos
α
sin
z
,
T
u
ρ
=
cos
z
.
T
u
for the
elevation angle
h
. The above formulas are the basis for the
M
-file
topocent
.
The angle
h
or rather sin
h
is an important parameter for any procedure calcu-
lating the tropospheric delay, cf. the
M
-file
tropo
.
Furthermore, the quantity sin
h
has a decisive role in planning observations:
when is
h
larger than 15
◦
, say? Those are the satellites we prefer to use in GPS.
Many other computations and investigations involve this elevation angle
h
.
From this we determine
α
and
z
. Especially we have sin
h
=
cos
z
=
ρ
Often we want to transform topocentric coordinate differences
(
x
,
y
,
z
)
to local
. This transformation is achieved by
F
T
:
⎡
coordinates
(
e
,
n
,
u
)
⎤
F
T
⎡
⎤
e
n
u
x
y
z
⎣
⎦
=
⎣
⎦
.
(8.43)
This transformation is implemented in the
M
-file
cart2utm
. Correspondingly, the
covariance matrix
enu
is given as
F
T
enu
=
x
F
.
(8.44)
Example 8.3
A receiver position is (3,435,470.80, 607,792.32, 5,321,592.38)
with the following covariance matrix, unit m
2
:
⎡
⎤
25
−
7
.
970
18
.
220
⎣
⎦
.
x
=
−
7
.
970
4
−
6
.
360
18
.
220
−
6
.
360
16
56
◦
55
,
10
◦
02
, and we get
We know the point has
ϕ
=
λ
=
⎡
⎤
−
0
.
1742
0
.
9847
0
F
T
⎣
⎦
;
=
−
0
.
8252
−
0
.
1460
0
.
5457
0
.
5374
0
.
0951
0
.
8380
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