Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 20

completing the square for r 0 + h and taking the square root as

= (r 0 + h) 1 −

1 / 2

2 hr 0 ( 1 − cos D 0 )
(r 0 +

r = [ (r 0 + h) 2 − 2 r 0 h( 1 − cos D 0 ) ] 1 / 2

h) 2

( 2 . 29 )

Since angle D 0 is very small, it can be approximated as

D 0
2

1 − cos D 0 ≈ ( 2 . 30 )

where D 0 is the angle expressed in radians. The r value can be written as

h) 1 −

h) 2

1 / 2

2 hr 0 D 0 / 2
(r 0 +

hr 0 D 0
2 (r 0 +

r ≈ (r 0 + = r 0 + h − ( 2 . 31 )

h)

At latitude of 45 degrees D 0 ( ≈ 1/297 radian) becomes maximum. If D 0 is
neglected, the result is

r 0 hD 0
2 (r 0 +

r ≈ r 0 + h −

h) ≈

r 0 + h ( 2 . 32 )

Using this result, if h r e = 6368 km (the average radius of

the earth), the error term calculated is less than 0.6 m. Thus

= 100 km, and r 0 =

h = r − r 0 ( 2 . 33 )

is a good approximation. However, in this equation the value of r 0 must be
evaluated, as discussed in Section 2.12.

2.12 CALCULATION OF GEODETIC LATITUDE ( 5-7 )

Referring to Figure 2.7, the relation between angles L and L c can be found from
the triangle OPC . From the simple geometry it can be seen that

L = L c + D ( 2 . 34 )

If the angle D can be found, the relation between L and L c can be obtained.
To find this angle, let us find the distance OC first. Combining Equations (2.24)
and (2.27), the following result is obtained:

AE
tan L =

b e sin β
tan L

OC = OE − CE = a e cos β − a e cos β −

e e ) ] a e e e cos β e e OE

= a e cos β [1 − ( 1 − = = (2.35)

where β is not shown in this figure but is shown in Figure 2.6.