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