Differential GPS - Understanding GPS: Principles and Applications - page 382

(after application of the broadcast navigation data corrections) is in error by 10 ns, it
will result in a 3m pseudorange and carrier-phase measurement error for a user at
any location.
Since SA was discontinued in May 2000, satellite clock errors have var-
ied extremely slowly with time. Over short intervals (e.g., 1-60 seconds), a 1-2
mm/s rate of change is typical [1], corresponding to a satellite Allan deviation of
around 3 10 −12 s/s (see Appendix B for the definition of the Allan devia-

tion). Before SA was discontinued, satellite clock rate of change was limited by a
U.S. government commitment not to exceed 2 m/s in rate with a maximum accelera-
tion of 19 mm/s 2 [2]. The extremely rapid changes of SA-induced clock errors had a
very important influence on the operational DGPS systems that were developed
prior to 2000. As will be discussed in more detail in Section 8.3, many DGPS systems
provide estimates of first derivatives of pseudorange errors. This feature was driven
mostly by SA. Residual pseudorange errors (in meters) after applying pseudorange
rate corrections are on the order of 1/2 at 2 , where a is the acceleration of the error
(in m/s 2 ) and t is the latency of the correction (in seconds). For instance, with SA on
(i.e., a

× 10 −12 to 6 ×

0.019 m/s 2 ), pseudorange corrections had to be sent once per 10 seconds
to keep range errors due to latency less than 1m. This observation directly influ-
enced the DGPS data link requirements for many of the operating systems discussed
in Section 8.6.

=

8.2.2 Ephemeris Errors
As discussed in Section 7.2.2, errors in the broadcast satellite positions lead to
pseudorange and carrier-phase errors. Since the magnitude of ephemeris-induced
pseudorange or carrier-phase errors are dependent on the LOS between the user and
the satellite, these errors change with user location. However, the difference in
pseudorange or carrier-phase errors as seen by receivers in close proximity is very
small, since their respective LOSs to each satellite are very similar. To quantify the
amount of change, let the separation between a user U and reference station M be
denoted as p (see Figure 8.2). We will refer to the actual orbital satellite position as
the true position. The error in the estimated satellite position (i.e., the broadcast
ephemeris) is represented as d m be the true and estimated distances,

respectively, of the reference station to the satellite, and let d u and

ε S . Let d m and ′

d u be the corre- ′

sponding distances of the user to the satellite. Let φ m be the angle formed by the

directions of the reference station to the user and to the actual satellite position. Let
α

be the angle formed by the directions of the reference station to the actual and esti-
mated positions of the satellite, S and S ′ , respectively. The law of cosines gives us the

following two relationships:

( )

d ′

2

=′ d

2

+ p

2

− 2 pd cos φα −′

u m m m

ddp d

2

=+−

2 2

2 cos φ

u m m m

in elevation angles between the actual and esti-
mated satellite positions from the monitor station. (The absolute value of

where α ′ is the difference φφ

mm

− ′

α ′ is less

than or equal to the absolute value of α , and the two are equal when the two trian-

gles lie in the same plane.)