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International cooperation and technology transfer
Mussio, Luigi

are equal. Any interpolator would give a constant
geoid value even under the mountain. Obviously,
this is wrong since the geoidal signal due to the
mountain is completely missed. On the contrary, by
introducing the N rt c component, this signal is
correctly modelled and taken into account. To
quantify the mismodelling that can be introduced by
an improper handling of the topography, a very
simple example can be shown.
Two parallelepipeda are combined to simulate the
geoidal effect of a mountain and the undulation
values are supposed to be sampled in points A and
B (see Figure 2).
Figure 2. The simulated mountain
By assuming a density of 2.67 g/cm**3 and using
the closed formula giving the potential of a
parallelepiped (W.D. Mac Millan, 1958), a common
value of 10 cm can be computed for the terrain
component N rtc . Hence, any interpolator would give
the same value in C, placed on top of the mountain,
while the correct directly computed value is 16 cm.
So, a bias of 6 cm is introduced by disregarding the
topographic signal, even considering a quite small
Thus, this simple example proves that with an
unhomogeneous distribution of data, biased
estimations can occur indipendently from the used
In the following, numerical tests will be carried out to
define the optimal procedure and the best
interpolation method for geoid prediction from
GPS/levelling derived undulations.
2.1 The geoid simulation procedure
Simulated geoid data in selected areas have been
computed according to the following scheme:
1) geoid values on a regular 3’ grid;
2) scattered geoid values (point positions as
derived from real data distributions);
3) geoid values on a regular 5’ grid.
The low frequency component has been derived
from the OSU91A geopotential model (R. Rapp,
1997), the RTC effect has been computed by using
the Italian DTM data base (M.T. Carrozzo et
al,1982) and the residual geoid signal N r has been
obtained from the ITALGE095 solution (R. Barzaghi
et al.,1996). For each selected area, two different
interpolation methods have been applied to predict
from data sets 1) and 2) on 3). So, data set 3) is the
target data set while 1) and 2) are the input data
bases. Using data base 1) to predict on 3) is quite
unrealistic, since, as it is pointed out before, real
data are not usually regularly distributed.
Nevertheless, this computation allows testing of
different interpolation methods in an optimal input
data case. On the contrary, prediction from 2) on 3)
presents a more realistic situation, where data
distribution is critical in determining the estimated
For both cases, predictions have been computed
a) the total geoid signal
N T =N M +N rtc +N r
b) the geoid data after low frequency component
N r =N t -N m
c) the residual geoid obtained by removing both
low and high frequency geoid signals
N r =N T -N M -N rtc .
This is done to account not only for interpolation
method and input data distribution on the final result
but also for spectral properties of input geoidal data.
2.2 Test area (A)
The area centered on the Trento - Bolzano
provinces has been selected as the first test area.
Data were considered in a window with boundaries
45.5 <(p <47 10.2 < A < 11.7
This is a rough topography area in the eastern Alps;
in Figure 3 the total geoid signal is plotted and the