(12)
, Guerra and
) » Cx and Cy
be applied if
own. On the
€ not exactly
astic values
ly useful for
make it more
| cy are vay
linates of the
while c, ani
7 the nominal
/ the photo
y ellipses is
he coefficient
ce it refers to
ecision.
tions
observational
, represents à
tion of modd
iod" (Kran,
weight p; à
dardized resi
(13)
iterations and
;nal matrix ()
articural cate
'cal model (8)
xr linearizable
| the approxi
y far from the
yaragraph 33.
first iterations
and for this
ntain also the
rst order, the
tionship (0
nction of the
To solve this problem a constant diagonal matrix Q = yl, where
= 0.1, is applied for the first 7 iterations, and the danish
method is applied for the following ones.
33. Numerical problems
The analitycal model reported in (8) makes it possible to solve
simultaneously two different problems: i.e. the cartographic
georeference and the non-conventional photogrammetric survey.
Furthermore for each phase a sort of specific problems are also
solved. These are the plane transformations between the
different map units and map layers and, for the photogram-
metric survey, the updating of particulars not already reported
in cartography. From the computational point of view a big
difficulty is given by the fact that for the most of the unknown
values to be estimated or predicted their approximative values
are not available.
For this purpose and for each group of unknowns contained in
the subvectors X4, X2, X3, X4 and s different operational
strategies have been identified.
The working method followed considers the geometrical
meaning of the different unknowns, and consists of a cascade
process for the computation of the approximate values.
The estimation of the parameters of an homologous
transformation and the planimetric coordinates of the carto-
graphic tie points, contained in the subvector x,, allows to
estimate the planimetric coordinates of the cartographic-
photogrammetric tie points X4. These values together with the
orthometric height x, make it possible to compute the
approximate values of the DLT parameteres contained in the
subvector X2. .
The increments of coordinates contained in the vector s and
relative to cartographic and photogrammetric control points
refer to approximate values which are more close to reality
because they could come, for instance, from GPS measure-
ments.
3.4. Use of GPS in cartography
Since the beginning, GPS has also been tried to determine the
coordinates of the photogrammetric check points (Ackermann,
1991). For this purpose it is useful to recall some of the aspects
jointed to the instrumentation, the errors of the methods, the
productivity and the cartographic applications.
The instruments available today are technically better than
those of some years ago, the receivers have a better signal to
noise ratio and are less sensitive to systematic errors. This has
permitted the obtaining of greater and greater precision,
especially in the differential use of the code. Furthermore, the
completion of the constellation has permitted, with a single
code, that is, with receivers that cost of a few thousands dollars,
the obtaining of sub metrical precision. These precisions are
already sufficient for the support and updating of middle and
small scale maps (1:10.000--1:100.000) but not yet for maps of
larger scale.
The use of the single code has the advantage of furnishing low
cost coordinates that are free of momentary losses of signal,
however, long sessions of measurements are necessary to obtain
differential sub metrical precision because of multi-path errors
(Yola and Kleusberg, 1991).
The precision and the productivity increases with the use of
single frequency receivers which, at the moment, have a cost
that can be compared with that of a good total station. It is not
necessary to use two frequencies for the support or the
construction of large scale maps (1:1.000+1:5.000) up to a
197
distance of 10-15 km. In this extent the precision furnished by
the construction companies varies from +(lcm+2ppm) for the
static modality to +(2cm+2ppm) for the stop-and-go method.
The most productive methods, in increasing order, are: the
pseudo-static method which forces the reoccupation of site after
about an hour, the stop-and-go method and the continuous
kinematic method.
The kinematic method can also be used for positioning in real
time (RTK) with the use of transmitting station, a receiver and
two radio modem (Allison, Griffioen and Talbot, 1994).
The stop-and-go method forces the presence of a good
constellation (at least 5 satellites). All the methods are suitable
for a maximum distance of 10+15 km.
The use of the single L1 frequency introduces "iono-
deformations" on a ground network, which as a first approxi-
mation, are assimilable to a scale factor, even though it is
possible to recover 90% of the ionospheric delay with the diver-
gence method in the C/A & L1 receivers.
The use of the predicted ephemeris, which are indispensable for
the RTK positioning technique, causes an error which is
reflected on the baseline components, but only slightly on the
slope distances.
These and other errors increases for long distances of the fixed
receiver, then is not possible to use data from a fixed station
national service over the 15 km distance, but instead it is
necessary to use a couple of L1 receivers.
3.5. Transformation problems
GPS cartographic surveying permits updating of a GIS database
with new points or obtain the GPS coordinates from the already
surveyed points. In this second case the proposed technique
allows one both to obtain the transformation parameters from
among the GPS coordinates, cartographic coordinates and
orthometric heights and to improve the global precision of the
map. This latter requirement is very useful if one uses the
digital version of cartographic products or cadastral maps.
It is known that the precision for long distances is poor for
these maps, while instead in this case the GPS system furnishes
the greatest precision.
To obtain the (@,À,h), coordinates of a set of points referring
to a local ellipsoid (n) by means of the (q,A,h),, of the same
points referring to another ellipsoid (w) that is differently
oriented, the well-known Molodensky formula is useful. Let's
imagine that the first system is the national system, in which
the cartography is referred, and the WGS84 geocentric system
is the second. It is known in Italy, from experience, that the
variations due to passage are of the order 100-200 m in
planimetry and of about 50 m in altimetry. One can write
(Pierozzi, 1989):
ep
A, 17] A, | - A($X9,9Y9,6Z9, 86,96, ,067,9a,00)! — ABT
h h
n w
(14)
The first six components of the 8 vector are the shift and
rotation between the two systems, the last two are the semiaxis
and the ellipsoidal flattering variations. The hypothesis that no
scale factors exist between the systems has been assumed in the
formula. The terms of matrix A are known non-linear functions
of (@,A,h),, the & terms are "a priori" unknowns of the
problem (apart from terms da and da). When, in a certain area,
a certain number of points are available in both systems,
equation (14) is useful to obtain by a least square solution these
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
