62 
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 
estimate. 
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 
components 
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 
available 
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 
residuals 
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 
areas. 
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 
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2. 
2.1 The ge 
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