International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 
   
  
  
  
D eee etat D ^ 
^ d ou d dà ?" 3 10 
dus 8 7 8 
v d 
2 
Figure 2. Design of the convergent geometry 
  
Figure 3. Correlated clouds of points. 
3. METHODS 
Different estimates have been given, all of them validated 
afterwards, letting us know the mean square error and compare 
models' accuracy. 
3.1 Inverse Distance Weighing (IDW) 
The z coordinates of the point to interpolate are estimated 
allocating weights to environment data in inverse relation to 
distance; nearest points thus getting more weight in 
calculations. It is an exact method that estimates the value of 
the variable for a point not belonging to the sample, using the 
following expression (1): 
2 
i 
i=l 
zi 
> = 
ON 
| 
i=l 
where dj; is the Euclidean distance between each data item and 
the point to interpolate, and p is the weighting exponent. The 
least mean square error of the prediction (RMSPE) is calculated 
in order to determine the optimal exponent value. The optimal 
power (p) value is determined by minimizing the root mean 
square prediction error (RMSPE). The RMSPE is the statistic 
that is calculated from cross-validation. In cross-validation, 
each measured point is removed and compared to the predicted 
value for that location. The RMSPE is a summary statistic 
quantifying the error of the prediction surface (Johnston et.alt. 
RMSPE 
M Op value 
  
1.5 2 2.5 
exponent 
e 
Un 
Lo 
Figure 4. Optimal exponent determination graph. 
3.2 Radial basis function (RBF) 
Radial base functions comprise a wide group of exact and local 
interpolators that use an equation with its base dependent on 
distance. Generally speaking, the value of the variable is given 
by the following expression (2): 
z; = NO ; F(d;) (2) 
i=l 
where F(dij) is the radial base function, with d being the 
distance between points; a, the coefficients that will be 
calculated solving a lincar system of n equations, and n, the 
number of neighbouring sample points involved in obtaining z;. 
In this case, we will use a radial base multiquadratic-type 
function (3)(Aguilar et. al, 2001), which comprises an r 
parameter: the softening factor. This value should be previously 
tested according to the data in each case; a very high value will 
generate a very softened surface, far from the real surface. 
F(d;) 24d; *r' (3) 
, Vol XXXV, Part BS. Istanbul 2004 
   
   
  
  
   
  
  
   
  
   
   
   
   
  
  
  
  
  
   
  
  
  
  
  
  
  
    
  
    
   
   
  
   
  
  
  
   
  
   
  
   
   
   
  
    
  
  
   
   
   
     
  
Internatioi 
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