Part B5. Istanbul 2004
en each data item and
ighting exponent. The
RMSPE) is calculated
it value. The optimal
nizing the root mean
MSPE is the statistic
. In cross-validation,
pared to the predicted
; a summary statistic
face (Johnston et.alt.
nination graph.
oup of exact and local
its base dependent on
f the variable is given
Ó with d being the
ficients that will be
equations, and n, the
/olved in obtaining z;.
c multiquadratic-type
hich comprises an r
e should be previously
a very high value will
the real surface.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
3.3 Kriging
It is an exact and local interpolation method ( Moral, 2003) that
sets the weight of each sample point according to the distance
between the point to interpolate and the sample points.
Kriging's procedure estimates this dependence over the
semivariance, which takes different values according to the
distance between data items. The function that relates
semivariance to distance is called semivariogram and shows the
variation in correlation among the data, according to distance.
The basic expression is (4):
| € )
HA) = — Zim Zine) (4
7(h) à 1) )
i=l
where n is the number of value pairs separated by a distance h.
Theory demands the semivariogram to be of general validity for
the whole digital model's area. This means that data
interdependence should be the exclusive function of the
distance among them, and not of its absolute space location,
because of which it doesn't allow for the treatment of
discontinuities that lead to abrupt changes, such as slope
ruptures.
4. RESULTS
To evaluate the effectiveness of the interpolation methods, the
data were validated through a random mesh of points (figure 5),
whose coordinates were measured using multiple direct
intersection by classic topography.
=
Figure 5. Distribution of the validation points.
For the statistical evaluation of the effectiveness of each
method, the model we obtained was checked with the real
model, using the root mean square error (RMS) over the 72
validation points, which is defined by the following
expresion(5) (Felicísimo, 1994):
n
( _ estimated _ real )
7 — 7.
ei fni
RMS 2 |
er (
n
un
—
on X^
co o
C» un
A
Predicted, 10-2
MN) No M) M) NN
nn
eoo
M
On
OY
2,55 ; 2, 57 ; 2,58 2,58
2
Measured, 10-
Regression function. 1,001 * x + -0,362
Figure 6. Result of points’ validation
There are different search types among which we must choose
to select those neighbouring sample points that will take part in
the numeric determination of the "non-sample" point. This can
be performed taking quadrants, octantes or the whole circular
sector into account. So as to research the influence we
performed the IDW interpolation using the three types of
neighbour selection for 30 neighbours of which at least 12
within a search circumference of a 10cm radius, obtaining the
following final RMS (table 7):
Selection RMS (m)
all 0.025
quadrants 0.028
octantes 0.030
Table 7. Model's error according to different search types by
Sectors.
4.1 Optimal number of neighbouring points
The number of neighbouring points that take part in thc
interpolation was calculated evaluating the RMS, using IDW as
a method.The method without quadrants has been selected for
being the one with best results, as seen in the previous section.
Number of Minimum number of RMS (m)
neighbours neighbours
45 15 0,026
30 I? 0.025
15 10 0.026
8 8 0.032
6 4 0.033
Table 8. RMS according to the number of neighbours selected
According to the table 8, there is a certain threshold above
which the interpolated model's precision doesn't improve, no
matter how high the number of points considered. For this
reason, the options of 30 neighbours with at least 12 within and
that of 15 neighbours with at least 10 are considered valid, since
having more points without improvement in precision only
increases the volume and time of calculations.
