Global Positioning System Reference
In-Depth Information
where
e
p
2
2
e
p
sin 2
L
sin
2
L
1
=−
sin
L
+
+···≤
1
.
6arc
−
sec
(
2
.
46
)
Substitute the approximation of
D
0
≈
e
p
sin 2
L
into Equation (2.43) and the
result is
D
=
2
e
p
1
−
2
1
−
1
2
sin 2
L
+
e
p
sin 2
L
sin
2
L
e
p
h
r
0
(
2
.
47
)
or
D
=
e
p
sin 2
L
+
(
2
.
48
)
where
e
p
2
he
p
r
0
=−
sin 2
L
−
sin 2
L
+···
(
2
.
49
)
This error is less than 4.5 arc-sec for
h
=
30 km. Using the approximate value
of
D
, the relation between angle
L
and
L
c
can be found from Equation (2.34) as
L
=
L
c
+
e
p
sin 2
L
(
2
.
50
)
This is a nonlinear equation that can be solved through the iteration method. This
equation can be written in a slightly different form as
L
i
+
1
=
L
c
+
e
p
sin 2
L
i
(
2
.
51
)
where
i
L
i
)
is smaller than a predetermined threshold, the last value of
L
i
can be considered
as the desired one. It should be noted that during the iteration method
L
c
is a
constant that is obtained from Equation (2.18).
=
0
,
1
,
2
,...
. One can start with
L
0
=
L
c
. If the difference
(L
i
+
1
−
2.13 CALCULATION OF A POINT ON THE SURFACE OF THE EARTH
(
5
)
The final step of this calculation is to find the value
r
0
in Equation (2.33). This
value is also shown in Figure 2.7. The point
A
(
x
,
z
) is on the ellipse; therefore,
it satisfies the following elliptic Equation (2.21). This equation is rewritten here
for convenience,
x
2
a
e
+
z
2
b
e
=
1
(
2
.
52
)
where
a
e
and
b
e
are the semi-major and semi-minor axes of the earth. From
Figure 2.7, the
x
and
z
values can be written as
=
x
r
0
cos
L
co
z
=
r
0
sin
L
co
(2.53)
Search WWH ::
Custom Search