Differential GPS - Understanding GPS: Principles and Applications - page 404

Clearly, we can incorporate this result into the DD as well:

( )

pq pq − 1 pq pq pq

DD =⋅ be λ = φ + N + S (8.14)

km km km km

where b · e pq is the inner product between the unknown baseline vector and the dif-
ference of the unit vectors to SVs p and q . Since determining the unknown baseline
between the antennas is at the heart of the matter, it is this second formulation for
the DDs, (8.14), that will serve as the basis for further derivation.
Of the variables shown in (8.14), only one can be precisely measured by the
receiver—the carrier phase. In actuality, then, it is the carrier-phase measurements
of the receivers that are combined to produce the DDs. The term DD cp is adopted
to represent this, and implicit in its formulation is conversion to meters. The noise
term will be dropped to simplify the expression. In the end, as the carrier-cycle
ambiguity search progresses, the noise sources tend to cancel. There remains to be
determined the baseline vector ( b ), which has three components ( b x , b y , b z ), plus an
unknown integer carrier-cycle ambiguity ( N ) associated with each of the DD cp
terms. Toward this end, four DDs will be used. While additional DDs could be
formed depending on the number of satellites in track by the receiver, this is a suffi-
cient number and minimizes the computational requirements of the carrier-cycle
ambiguity-search algorithm. In terms of satellites, two satellites are required to
form each DD. Thus, in order to form four DD equations, a minimum of five satel-
lites is necessary. The transfiguration and extension of (8.14) to four DDs appears
as follows:

DD
DD
DD
DD

e e e

N
N
N
N

cp 1 12 x 12 y 12 z 1

b
b
b

e e e

x

cp 2 13 x 13 y 13 z 2

= + λ (8.15)

e e e

y

cp 3 14 x 14 y 14 z 3

e e e

z

cp 4 15 x 15 y 15 z 4

where DD cp 1 , for example, is the first of four DDs, e 12 represents the differenced unit
vector between the two satellites under consideration, b is the baseline vector, N 1 is
the associated integer carrier-cycle ambiguity, and

is the applicable wavelength.
The wavelength is introduced at this point to provide consistency with DD cp and b ,
which are now in meters. During this and subsequent discussion, all DD formula-
tions will be in units of length. Using matrix notation, (8.15) takes the following
form:

λ

DD =+λ Hb N (8.16)

cp

3 data matrix
containing the differenced unit vectors between the two satellites represented in the
corresponding DD, b is a 3

where DD cp is a 4 × 1 column matrix of carrier-phase DDs, H is a 4 ×

× 1 column matrix of the baseline coordinates, and N is a

4 1 column matrix of integer ambiguities. Once the carrier-phase DDs are formed,

a similar set of DDs is determined using the pseudoranges between each antenna and
the same set of satellites.

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