Differential GPS - Understanding GPS: Principles and Applications - page 410

few integer wavelengths depending on the effects of satellite geometry and the sever-
ity of the multipath environment surrounding the antenna at either end of the
baseline.
Using the vector notation introduced with (8.19), the DD baseline equation for
the smoothed-code DDs is as follows:

DD = Hb (8.24)

s float

1
SVs, H is an m 3 data matrix containing the differenced unit vectors between the
two SVs represented in the corresponding DD, and b is a 3

In a general least-squares sense, DD s is an m ×1columnmatrixofDDsfor m +

1columnmatrixofthe
estimated floating baseline solution coordinates. Were the least squares solution for b
the only desired result, the generalized inverse approach H T H could be applied imme-
diately. In this situation, however, the floating baseline solution represents an inter-
mediate step along the way to the desired final result, which is an integer-ambiguity
resolution or the fixed baseline solution. With this end in mind, some matrix condi-
tioning is performed on the elements of (8.24) prior to determining the floating base-
line solution. The H matrix is decomposed using QR factorization ,where Q is a real,
orthonormal matrix (thus Q T Q

×

I )and R is an upper triangular matrix [29]. QR
factorization allows the least squares residual vector to be obtained by projecting the
DDs onto a measurement space that is orthogonal to the least-squares solution space
spanned by the columns of H . Hence, the least-squares residual vector is projected
onto the left null space of H , called parity space , while the least-squares solution is
mapped onto the column space of H , known as the estimation space [26]. Since the
parity space and the estimation space are orthogonal, the residuals therein are inde-
pendent of the estimate. This will be used to an advantage to isolate the carrier-cycle
integer ambiguities and subsequently adjust the smoothed-code DDs. Incorporating
the properties of the QR factorization into (8.24) yields:

=

DD = QRb (8.25)

s float

Capitalizing on the property of the orthonormal matrix, where the inverse and
transpose are equivalent, and then rearranging gives:

Rb = Q DD

T

(8.26)

float s

Expanding the matrices for clarity yields:

DD
DD
DD
DD

RR R
RR
R

QQQQ
QQQQ
QQQQ

T T T T

s

11 12 13

b
b
b

11 12 13 14

1

x

0
00
000

T T T T

s

22 23

=

21 22 23 24

(8.27)

2

y

T T T T

s

33

31 32 33 34

3

z

qq qq

s

1 2 3 4

4

Equation (8.27) lends itself readily to horizontal partitioning, and elements of
the Q T matrix have been labeled with capital Q to show the portion that corresponds
to the least-squares solution (estimation space) and with small q to indicate the ele-