NavCom SF-3050 A Computationally Efficient Ambiguity Resolution - Manual (Page 3)

3
) ( ) ( / ) (
2 1 2 1
r r i i
w
r i i
w
R R N
φ φ φ φ λ − + − − − =
(5)
where the superscript, i, represents each satellite in turn
and the superscript, r, represents the reference satellite—
in this case the highest elevation satellite.
Our initial wide-lane carrier-phase RTK solution involves
rounding the floating ambiguities so obtained to their
integer values, substituting these values into equation (4)
and computing a new least-squares position solution. The
floating ambiguity given by equation (5) is used in the
search process to sequentially select the closest integer
values over the selected search range.
We have found that searching over the four closest values
is almost always sufficient to include the true ambiguity
value. Given n satellites this means that there will be 4
n-1
permutations of the ambiguity vector. Thus, there will be
1024 permutations when six satellites meet the
appropriate elevation criterion and 4096 when seven
satellites are available. With each ambiguity permutation
a set of wide-lane carrier-phase measurement residuals
can be computed. It is these residuals which play a
prominent role in determining the correct set of integer
ambiguity values. However, obtaining the residuals by re-
computing a least-squares solution for each permutation is
computationally prohibitive. A much faster way to
compute the residuals is to use a residual sensitivity
matrix, S.
RESIDUAL SENSITIVITY MATRIX
Given the linearized measurement equation:
z Hx =
(6)
where H is the sensitivity matrix (i.e. the direction cosines) of
the state vector (position and clock corrections), x, to the
innovations (difference between the measurements and their
expected value), z.
Equation (6) can be expanded to include a set of position
and clock corrections corresponding to a set of
innovations:
Z HX =
(7)
Now if we want to see the separate effect on the position
of a single whole-cycle change in the ambiguity value of
each satellite, Z will become the identity matrix (or
depending on units, the identity matrix times the scalar
wavelength).
I HX =
(8)
The least-squares solution for X is then:
T T H H H X 1 ) ( − =
(9)
For a weighted least-squares solution this becomes:
1 1 1 ) ( − − − = R H H R H X T T
(10)
where R is the measurement covariance matrix.
Note that the left hand side of equation (9) or (10) can be
pre-computed and when multiplied by the appropriate
column of the identity matrix (or scaled identity matrix)
gives the appropriate column of X, which is the associated
change in the position and clock caused by the integer
ambiguity change.
Multiplying equation (9) by H tells us how much that
change in position will affect the innovations.
T T H H H H HX 1 ) ( − =
(11)
Now if we subtract this change in the innovations from
the input value of the innovations (I) we get the effect on
the residuals of a whole cycle change in the ambiguity
value for each satellite. This is called the residual
sensitivity matrix, S, and is given by:
T T H H H H I S 1 ) ( − − =
(12)
The residual sensitivity matrix, S, for a weighted least
squares solution is:
1 1 1 ) ( − − − − = R H H R H H I S T T
(13)
The S matrix has a number of interesting properties. It is
symmetric. It is idempotent, i.e. S=S
2 =S 3 =… The sum of
any row or column equals zero, i.e. residuals are zero
mean. The length of any row or column is equal to the
square root of the associated diagonal element. Since the
solution vector, x, has four elements, the rank of S is n-4
where n is the number of satellites.
The residuals of the initial RTK solution, described in the
prior section are updated by adding the product of the S
matrix and the matrix formed by scaling the identity
matrix diagonal elements by the specific permutation of
changes in the integer ambiguity values to be tested. The
ten permutations with the smallest root-sum-square (rss)
of residuals are saved for further narrow-lane processing
if their rss residuals is less than an acceptable threshold
value.
STEPPING TO THE NARROW-LANE
Assuming the differential ionospheric refraction is
negligible allows us to equate the narrow-lane range to
the wide-lane range. Thus:
w w
N N λ φ φ λ φ ) ( ) (
2 1 1 1 1
− + = +
(14)
Solving equation (14) for the L
1
ambiguity value gives:
)
17
9
4
17
9
3 (
17
9
4
2 1 1
φ φ − + =
w
N N (15)
It is interesting to note that the deviation from an integer
value obtained for N
1
in equation (15) is identical in