Global Positioning System Reference
In-Depth Information
position estimate is not optimal. As derived in Appendix A, if the pseudorange
errors are Gaussian and the covariance of UEREs for the visible satellites is given by
the matrix R , then the optimal solution for user position is given by the WLS
estimate
(
)
1
xHRHHR
=
T
1
T
1
(7.67)
(Note that, as with the ordinary least-squares solution, we are truly solving for a
correction to an initial estimate of the user position and clock error.) Equation
(7.67) collapses to (7.33) in the case when R
2
n iden-
tity matrix, as expected since this case corresponds to our original independent and
identically distributed assumption. For a general matrix R, (7.67) can be thought of
as implementing an optimal weighting of pseudorange measurements based on their
relative noise levels and relative importance for each estimated quantity.
As one example of an error covariance matrix, consider the single-frequency
SPS user whose pseudorange measurement errors are dominated by residual iono-
spheric delays. As noted in the discussion in Section 7.3.2, residual ionospheric
errors for single-frequency users are highly correlated. The covariance matrix of
residual ionospheric errors can be approximated as
= σ UERE
I
with I equal to the n
×
()
()( )
()()
m
2
el
m el
m el
L
m el
m el
1
1
2
1
n
()()
()
mel mel
m
2
el
2
R
= σ iv
1
2
2
(7.68)
M
O
()
()
()
mel mel
m l
2
1
n
n
σ i 2 is the residual vertical ionospheric delay variance, which could be approx-
imated as some fraction of the Klobuchar vertical delay estimate. The ij th element of
the matrix in (7.68) is the product of two ionospheric mapping functions, m ( el ). For
example, (7.21) could be used, corresponding to the elevation angles ( el ) for satellite
i and j .
Another typical example of a covariance matrix is a diagonal matrix whose
diagonal elements are obtained using an approximation for pseudorange error vari-
ance versus elevation angle, usually a monotonically increasing function as eleva-
tion angle decreases (e.g., see [14]). The use of such a covariance matrix in a WLS
solution deweights low-elevation angle satellites that are expected to be noisier due
to typical characteristics of multipath and residual tropospheric errors.
where
7.3.4 Additional State Variables
Thus far, we have focused on estimation of the user's ( x , y , z ) position coordinates
and clock bias. The complete set of parameters that are estimated within a GPS
receiver, often referred to as the state or state vector , may include a number of other
variables. For instance, if in addition to pseudorange measurements, Doppler mea-
surements (from an FLL or PLL) or differenced carrier-phase measurements are
available, then velocity in each of the three coordinates (
&
t u ,
may also be estimated. The same least squares or WLS techniques used for position
xyz , and clock drift,
,
,
&
)
&
&
 
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