Differential GPS - Understanding GPS: Principles and Applications - page 383

ε s

S'

S

d m

d m

d u

d u

α

S - Satellite position
S' - Estimated satellite position
M - Reference station position
U - User position

φ m

φ m

M U

User

p

Reference
station

Figure 8.2 Variation of broadcast ephemeris errors with viewing angle.

d m , and neglecting
the higher order terms in the binomial expansion of the square root in each of these
equations, we obtain

Solving the first equation for d mu and the second for d u − ′ − ′

⋅

1
2

p
d

1
2

dd ′− ′≈− ⋅ pp +⋅ cos φα +′ ⋅ p ⋅ sin φ +⋅ α⋅⋅ ′

2

p cos φ

m u m m m

′

m

1
2

p
d

dd −≈+⋅ ⋅ pp −⋅ cos φ

u m m

′

m

Adding these two equations, we find that the difference between the errors,

ε u = dd ′ − and ε mmm = dd ′ − , is

u u

1
2

( ) ( )

ε −=′ −′ ε dd + d d − =′ ⋅ α p ⋅ sin φ +⋅ α ′

2

⋅ p ⋅ cos φ

mu u m m u m m

or

1
2

( ) ( )

2

ε − ε = dd ′−′ + d d − ≤ ⋅⋅ α p sin φ + ⋅ α ⋅⋅ p cos φ

mu u m m u m m

where the equality holds if the estimated satellite position lies in the plane defined
by the user position, reference station position, and true satellite position.
The difference

is the error introduced by the pseudorange correction at
the user. To simplify the expression, assume that the angle

εε
mu

−

φ m is greater than 5º, that
the separatio n be tween the user and referenc e sta tion is less than 1,000 km, and that
the direction SS ′ is parallel to the direction MU . Then

ε ⋅ sin φ

=

ε

S m S

sin 2 ε −≤⋅ ε α p ⋅ sin φ ≈ ⋅⋅ p sin φ ⋅⋅ p φ (8.1)

mu m m m

d d

m m