the 
same 
ance 
is 
| 
| 
| 
e of | 
| 
| 
| 
| 
| 
| 
the 
re no 
    
where e = s- s 
p P 
At the check points, if any, discrepances can be 
computed by: 
where Ww are a small set of data use for the 
crossvalidation. 
2.3 Finite element method (e.g. tricubic spline 
interpolation) 
A tricubic spline function is given by the product 
of three orthogonal cubic spline functions: 
S(x,y,z) 9» S(x) Sly) S(z) 
The choice for the number of cells and the number 
of knots depends on the number of observations m 
and the interpolation step ô. 
The number of cells is the product of the number 
of classes in three directions x,y and z: 
x uy az 
where p int (AX/8) +.1 
um z int (AY/8) * 1 
v - int (42/8) + | 
z 
being AX, AY and AZ the dimensions of the space 
region in three directions and & the chosen 
interpolation step. Consequently the number of 
knots is: 
n-n5n n n sp 43)4(» 43] (y^ € 3) 
X-:y--Z x y zZ 
The tricubic spline interpolation is performed, as 
a classical least squares problem, by writing a 
system of observation equations: 
Hays k = 1,m 
k 
4 
4 
Lo. e pau Ji j L' $ 1 (5 1, Sy) 
s = 
k 
D» =" 
i=1 
and associating it with the least squares norm: 
  
The weights are mostly assumed equal one; however 
more complex stochastic model should be defined 
including correlations between the observations, 
but they are usually omitted in sake of 
brevity. 
The following formulas are the legenda of the 
functional model; indeed for the x direction, the 
coordinate of the k-th knot respect the initial 
corner is splitted in two parts: 
Ax =1 8 + 8x 
k k 
where the number of the preceding knots is: 
I = int (Ax /8) 
and the position inside the class is: 
Er = 8x /ó 
being 8x = Ax - IS 
k k 
analogously, for the y direction: 
Ay, =J 3d + y. 
where J = int (Ay, /8) 
and "n ay /9 
being ôy, = Ay, — I8 
and for the z direction: 
Az L6 t 62 
k k 
where L - int (Az 78) 
and C - óz /ó 
k k 
being óz, - Az - Ió 
k k 
Note that suitable constraints for the knots 
should be introduced at the border and in empty 
regions. 
3. THE SYSTEM OF PROGRAMS 
(Because of the modularity of the system of 
programs, there is a high degree of similarity 
between this system and those dedicated to 2D 
problems and time series, see: Crippa/Mussio, 
1987. ) 
The system consists of a set of programs, which 
allow for the following operations: