ST PA AE EEE 
SR RT RR aa 
  
w=- ÿ >=. y k 2 1,m 
A criterion for the choice of the radius of the 
sphere including the first autocovariance zone is 
maximizing the first autocovariance estimate as 
follows: 
(1), | eo] = | e 
r : ylr = max |vir 
,U) 
n i 
| A 531 1 (1) 
where y|r = = Vv — V 
n id-1 ‘+ a ) j=1 À 
1 
and Ko, V P, a9 «1l P. = Pl < pat 
The "best fit" of the autocovariance function of 
the signal is then chosen among Some available 
models, namely: 
E y(r) exp(-br) 
Il 
g 
N y(r) = a exp(-br^) 
EP y(r) » a exp(-br) (1-cr?) 
NP y(r) = a exp(-br^) (1-cr?) 
ES y(r) = a exp(-br) sin(cr)/(cr) 
NS y(r) = a exp(-br^) sin(er)/(er) 
EJ y(r) = 2a exp(-br) J (er)/(cr) 
NJ y(r) » 2a exp(-br^) J (er)/(cr) 
where the smoothness given by the coefficient b 
for the cases EP and NP is very high. The previous 
abbreviations indicate respectively: 
E exponential function; 
N normal function; 
P parabole function; 
S sine function over X; 
jJ Bessel function of 1st order over x. 
This list has been built according to the 
definition of covariance function: positive power, 
i.e. positive 3D Fourier transform, and Schwarz 
condition for vectorial processes. New covariance 
function can be created from old by applying 
following fundamental theorems: 
- a linear combination with positive coefficients; 
- & product; 
- & convolution. 
The same list is used to interpolate 
crosscovariance functions: it is not correct in 
principle, but is acceptable in practice, provided 
that crosscovariance estimates are low enough. 
Besides, since 3D isotropic finite covariance 
     
functions are not known, a 3D finite covariance 
function, isotropic from the numerical point of 
view, can be found using a tricubic  spline 
function: 
y(r) = S(x,y,z) = S(x) S(y) S(z) 
Finally, the noise variance is found as: 
2 2 2 2 
0 =6"-6-=-c0"r 4 
n s 
and the noise covariance can be found with a 
similar formula. 
2.2 Filtering, prediction and crossvalidation 
By using an hybrid norm: 
RE AS 7 De 2 t; = 3 o : 
óc stc +n no sal. grn=yv ) = min 
ss n 
the residuals v^ can be split in two parts: the 
signal s and the noise n: 
^ -1 o o 
eec C v.zNV ch 
ss vv 
neg Lgs cct v 
n vv 
were C =C +0 I 
vv ss 
and C = [y(r)] is the matrix of autocovariance of 
the signal, I is the unitary matrix of the same 
dimension as C 
As regard the Säccuracy, the variance-covariance 
matrix of the error of the estimated signal is 
given by: 
-o1-C- 
n nn 
i.e. for the main diagonal elements: 
o mo -c C o. e e* diag(C ^) 
n n vv 
where e =s — S 
and Crr= o* C 
n 
while a posteriori an estimate of variance of the 
noise is supplied by: 
VV 
dhs (Fc) 
The same relationships are employed for the 
prediction of the signal s in points where no 
observations are generally available: