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Figure 1: Satellite radial position and clock errors
(Entire constellation, GPS week 1217)
implies that the satellite positions could be weighted in the
adjustment instead of being fixed, just as the GPS exposure
station position observations are weighted in contemporary
GPScontrolled aerial photogrammetry.
32 Satellite Clock Biases
A satellite clock bias will manifest itself entirely as a range
error. Most of the satellite clock biases can be removed us-
ing correction coefficients broadcast as part of the satellite
ephemeris. The residual error that remains, however, can
still be significant. Figure 1(b) shows the difference be-
tween the broadcast and precise satellite clock corrections.
When compared with Figure 1(a), it can be seen that the
residual satellite clock error is larger than the radial satel-
lite orbit errors. For the one week period shown in the
figure, the RMS clock error for the entire constellation was
just under 2m, and the maximum error was close to 10m.
To correct for the residual satellite clock errors, precise
clock corrections can be used. Such corrections are nor-
mally included with precise ephemerides, and, just as with
precise ephemerides, polynomial interpolation can be used
to determine the correction between sample epochs. Be-
cause the satellite clocks are very stable, a lower-order in-
terpolator can be used than that for the positions. However,
regardless of the interpolator order, with 15-minute clock
corrections maximum errors of close to a metre may still
occur. Fortunately, clock corrections at a 5-minute sample
interval are also available from the IGS, and when these
higher-rate corrections are used with a third-order interpo-
lator, the maximum errors can be reduced to under half a
metre.
3.3 Ionospheric Delays
Essentially, the only option for dealing with the ionosphe-
ric delays is to use the ionospheric-free linear combination
935
to eliminate the first-order effects. The only other conve-
nient option is to use the broadcast ionospheric prediction
model to estimate its effect. However, the broadcast model
only removes 5096-6095 of the error, and can leave maxi-
mum errors of some tens of meters.
Figure 2 shows the noise in the ionospheric-free linear com-
bination, as determined by differencing the code and phase
measured ionospheric error and removing the mean dif-
ference. When such a difference is performed, all com-
mon errors are eliminated, and what remains is predomi-
nantly code multipath and receiver noise. For the satellite
depicted, the noise for the near-zenith measurement was
about 0.3m. As the elevation of the satellite decreases, the
noise increases, following a relationship that — until about
15? elevation — can roughly be described by
e(90°)
eps sin(e) ' o
where £(90?) is noise at the zenith angle, and c is the satel-
lite's elevation angle. This simple cosecant relationship
was used in this project to estimate the variances of the
ionospheric-free pseudorange observations.
W Measureq delay
-— Ô d, no 90 )/sin(e)
A RMS delay
lonospheric Delay (m)
id 29 30 40 50 60 70 ©
Elevation (degrees)
Figure 2: Measured code ionospheric delay noise
(SV 1, day 2 of GPS Week 1217)
3.4 Tropospheric Delays
The errors due to the tropospheric delay are typically miti-
gated using a combination of zenith-delay models and map-
ping functions. The tropospheric models may use surface
measurements of temperature, pressure, etc., or they may
use standard empirical values. In this project, the UNB2
tropospheric model was used in conjunction with the Niell
mapping function (Collins et al., 1996; Niell, 1996). When
the tropospheric delay is corrected for using a model, a
residual tropospheric delay (which is generally elevation-
dependant) will still remain. Any residual delay common
to all satellites will, however, be compensated for in the
estimate of the receiver clock offset.
4 TESTING AND RESULTS
The direct inclusion of the GPS pseudoranges in the bundle
adjustment was tested using a block of imagery captured
using a medium resolution digital camera with an image
size of 4096x4096 pixels. The block consisted of 42 im-
ages collected from 7 parallel flight lines at a flying height
of roughly 900m. Fifty-three well-distributed check points
