Global Positioning System Reference
In-Depth Information
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Throughout this and subsequent chapters, a superscript identifies the satellite,
and a subscript identifies the receiver. Lowercase letters
p
and
q
generally label the
satellites, whereas
k
and
m
refer to receivers. In relative positioning when two or
more receivers observe at the same time, we occasionally refer to the concept of a
base satellite and a base station. Usually the letter
p
will denote the base satellite
and
k
the base station. One may think of
p
and
q
as the pseudorandom noise (PRN)
numbers of the satellite or simply as sequential numbers identifying the satellites. The
L1 and L2 carrier frequencies are indicated by the subscripts 1 and 2, respectively.
In this notation we write the L1-pseudorange observation at station
k
to satellite
p
as
P
k,
1
. The respective notation for the L1-carrier phase in cycles is
ϕ
k,
1
. The
symbol
p
k,
1
refers to carrier phases scaled to distance. Occasionally we identify other
terms that relate to pseudoranges or carrier phases with subscripts
P
,
ϕ
, and
Φ
.For
example,
I
k,
2
,P
identifies the ionospheric delay of the L2-pseudorange measurement
from station
k
to satellite
p
.
In many cases, the superscripts and subscripts also indicate specific functions of
the observables. For example, if we label the epoch of the observations by
t
, the single
difference (SD) and double difference (DD) are defined as
Φ
[17
Lin
—
-
——
No
PgE
ϕ
km
(t)
ϕ
k
(t)
ϕ
m
(t)
=
−
(5.1)
ϕ
pq
ϕ
km
(t)
ϕ
km
(t)
km
(t)
=
−
(5.2)
Th
is notation does not use a comma between subscripts
k
and
m
and superscripts
p
and
q
. Notice that the subscript combinations
km
and
pq
as used here imply a
di
fferencing of “
k
-
m
” and “
p
-
q
” respectively. The triple difference (TD) refers to the
di
fference over time,
[17
km
t
2
,t
1
=
km
t
2
−
km
t
1
ϕ
pq
ϕ
pq
ϕ
pq
∆
(5.3)
Th
e between satellite difference (BSD) is identified by
ϕ
p
k
(t)
ϕ
k
(t)
ϕ
k
(t)
=
−
(5.4)
The receiver sets the epochs of observation internally. Without going into the inner
workings of receivers, suffice it to note from the users' point of view that receivers
make measurements of several satellites (usually all satellites in view) at the same
epoch. The output is available typically at the full second, half second, etc. The user
usually can set the rate of output. With this understanding, we will be able to simplify
notation at times by simply dropping the time label
t
.
5.1 PSEUDORANGES AND CARRIER PHASES
The pseudorange is related to the distance between the satellite and the receiver's
antenna, implied by the epochs of emission and reception of the codes. The trans-
mission (travel) time of the codes is measured by correlating identical PRN codes
