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International cooperation and technology transfer
Mussio, Luigi

for local applications by using GPS and spirit
leveling, is to get its estimate in one point only,
which is usually selected in a baricentric position,
and to assume that the geoid is equal to this value
in the whole area. This assumption is usually
considered true when dealing with small areas (e.g.
less than five km side) but may be completely wrong
even in flat areas. Even if no relevant topography is
present, geoid undulations can vary due to crustal
density anomalies.
Hence, by assuming a “flat” geoid on an area,
biased estimates can be obtained. So, it must be
stated that this kind of procedure, althogh extremely
fast, should be in any case avoided.
1.2 The geoid frequency components and the
interpolation procedure
A more refined procedure to get reliable geoid
estimates from interpolation is based both on the
choice of an efficient interpolator and on a frequency
component analysis of the geoidal signal.
Usually, in estimating the gravimetric geoid, the so
called “remove-restore” techniques is used. This is
basically a frequency decomposition of the
gravimetric and geoidal signals. First of all, a low
frequency component of gravity is removed from
data; this model gravity is commonly derived from
global geopotential model such as EGM96( F.G.
Lemoine et al., 1998). Then a Residual Terrain
Correction (RTC) (R. Forsberg, 1984) is applied to
remove the high frequency component related to the
high frequency behaviour of the topography. The
residual gravity values are then used to get the
residual geoid by means of e.g. Stokes formula.
Finally, the total geoidal signal is recovered by
adding to the residual geoid the RTC geoidal
component (high frequency part of N) and the global
geopotential model implied undulation (low
frequency component of N). The key point of this
methodology is this fequency component
decomposition which allows the application of
Stokes formula to a medium frequency signal, the
residual gravity. In this way, the irregular high
frequency component of gravity is filtered out when
removing the RTC effect and the low frequency part
is subtracted with the global geopotential model,
thus reducing the outer zone effect on the local
As said before, this is done in connection to the
gravimetric estimate of the geoid but it can be used
as well in interpolating pointwise N values coming
from GPS/leveling.
Any interpolator performs in a more afficient way if
the input signal is regular. By removing the high
frequency component of N related to the high
frequency part of the topography, a smoother N
signal is considered.
Furthermore, if the low fequency component is also
removed by using a suitable geopotential model, the
outer zone effect is considered and a more reliable
estimate is obtained. Hence, if it can be assumed
that the geoid undulation is decomposed into three
N T =N M +N rtc +N r ( 3 )
N M = geopotential model geoid,
N rtc = high frequency geoid component
N r = residual geoid signal
a possible procedure to get accurate estimates of N
could be the following:
i) computing pointwise undulation values at double
points, where GPS/leveling observations are
N T (P i ) = h(P i )-H(P i ) (4)
ii) computing residual geoid values
N r (Pj) = N T (Pj) - N M (Pj) - N rtc (Pj) (5)
iii) computing a suitable linear interpolation of these
Nr(Pk) = Ia ki N r (Pi) (6)
iv) restoring the model and the RTC geoid effects to
the interpolated values
N T (Pk) = N r (P k ) + N M (P k ) + N rtc (P k ) (7)
Hence, interpolation is performed on the residual
geoid N r only and low frequency and high frequency
component are separately modelled, removed and
then restored.
In this way better results are expected and the
simulations that follow will prove this.
Furthermore, in this way, also possible biases
connected to the position of the points where we
have the i) values are avoided.
Usually, leveling benchmarks are along roads; this
implies that undulations computed in i) are mainly
sampled along valleys, at least in rough topography
Now, as an example, suppose that GPS/leveling
values are taken on both side of a mountain and
that on these two sides the computed undulations
are equal
geoid vali
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