International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
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As in equation (1), this transformation requires the RY (t)
rotation matrix that aligns the camera axes to the mapping
space axes and the r¢ pg offset vector between the GPS
antenna and camera perspective centre.
By rearranging Equation (2), the reverse transformation is
found to be
r, = p [ M (rM = rips) F r&ps) : (3)
Elimination of the third equation yields a pair of modified
collinearity equations,
ri1( Xp — Xaps) t riz(Yp — YaPs)
+r13(Zp — Zops) + TGPS
Ip = - C—, ; (4a)
F rai(Xp — Xaps) * ra (Yp — Yaps)
+r33(Zp — Zaps) + zGPs
r21(Xp — Xaps) + r22(ŸP — YaPs)
+ras(Zp — Zaps) + yaps
és 23 (21 GPS) + YGPS (4b)
Ta(Xp—- Xa ps) +r32(Yp — Yaps)'
+r33(Zp I Zaps) + ZGPS
By examining Equation (4), it can be seen that the expo-
sure station positions are no longer explicitly present in the
collinearity equations, and that essentially, the GPS posi-
tions form the *base' ofthe equations. This has a number of
advantages. First, the GPS positions can be directly used as
the initial approximates in the linearised collinearity equa-
tions. Second, because the GPS positions are one of the
quantities being adjusted, the position measurements can
be directly used as parameter observations. In this case,
the parameter observation equation is
M ^M
0 = raps — "GPS: (5)
where PM S represents the current estimate of the posi-
tion during the adjustment. Adjusting the GPS positions di-
rectly also means that they are one of the quantities output
by the adjustment. This allows for easy comparison with
the input positions, which in turn simplifies the analysis of
the results. Finally, expressing the collinearity equations as
a function of the GPS positions means that the inclusion of
the raw GPS pseudorange and phase measurements in the
adjustment can be done with greater ease than would other-
wise be possible, as such measurements are also functions
of the GPS positions. This last point provides the motiva-
tion for the project under investigation in this paper.
2.3 Inclusion of the Pseudorange Measurements
in the Photogrammetric Adjustment
With the collinearity equations expressed by Equation (4),
inclusion ofthe GPS pseudorange measurements in the pho-
togrammatric adjustment is straightforward. Essentially,
simple observation equations with the form
p= Iraps/sv| T c^, (6)
are added to the adjustment, where p is the GPS pseudor-
ange measurement, c is the speed of light, and At, is the
receiver clock bias. This last term is added to the adjust-
ment as an unknown parameter.
934
Including the pseudorange measurement in the adjustment
should improve the mapping accuracy and, more impor-
tantly, reliability. Furthermore, it enables GPS data to be
used when less than four satellites are visible. While this
is not generally an issue for aerial mapping platforms, it
could be beneficial for terrestrial mobile mapping systems.
3 GPS ERRORS
Equation (6) assumes that the only error in the code pseu-
dorange measurement is random noise. In reality, of course,
this is not the case. There are a number of systematic er-
rors present in the observations, and when the largest of
these are accounted for the pseudorange observation equa-
tion more completely resembles
D= Iraps/sv + örsv| (7)
— eALw = Dip) + dione + di ropo- (8)
In the above equation, Óórsy is the error in the satellite co-
ordinates, A£,, is the satellite clock bias, d;,,,, is the iono-
spheric delay, and dopo 1 the tropospheric (or neutral at-
mosphere) delay. These error sources and the techniques
to mitigate them are well documented — see, for example,
Hofmann-Wellenhof et al. (1994). However, for complete-
ness these errors and the specific steps taken in this project
to mitigate them are detailed below.
3.1 Satellite Position Errors
The GPS satellite position errors are commonly divided
into along-track, across-track, and radial components. Due
to the great distance to the satellites, the former two error
components do not significantly project onto the measured
ranges. Thus, they can, effectively, be disregarded. The ra-
dial error, however, directly impacts ranges observed from
the satellites and must therefore be examined. Figure |
shows the radial satellite position errors for the entire GPS
constellation during a one-week period in May of 2003.
For this period, the root-mean-square (RMS) radial error is
just over 1.1m and the maximum error is close to 5m.
The satellite position error, radial or otherwise, is easily
corrected for using precise ephemerides. Precise ephem-
erides are observed or predicted orbits that are freely avail-
able from a number of organisations that include the United
States’ National Imagery and Mapping Agency (NIMA) and
the International GPS Service (1GS). In the case of the lat-
ter, several products are available, with accuracies ranging
from 25cm for predicted orbits to better than 5 cm for ob-
served orbits with a two-week latency. Either accuracy is
well below the expected accuracy of the pseudorange mea-
surements. Precise ephemerides typically have a sample
interval of 15 minutes. To determine satellite positions be-
tween samples, polynomial interpolation is normally used
(Hofmann-Wellenhof et al., 1994). A seventh-order inter-
polator is typically sufficient.
It should be noted that it is also possible to view the GPS
satellite position errors as errors in the control points defin-
ing the photogrammetric network datum, instead of group-
ing them with the other range errors as was done here. This
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