d 1 =
√
(x−x 1 )^2 + (y−y 1 )^2 + (z−z 1 )^2
d 2 =
√
(x−x 2 )^2 + (y−y 2 )^2 + (z−z 2 )^2
..
.
dn=
√
(x−xn)^2 + (y−yn)^2 + (z−zn)^2
(9 .2)
of a general point (x, y, z )from each of the known data points. It is then possible to
use these distances to construct the weights
w 1 =d 2 d 3 d 4 ···dn, w 2 =d 1 d 3 d 4... dn, w 3 =d 1 d 2 d 4... d n,... w n=d 1 d 2 ···dn− 1 (9 .3)
Note that to form the weight wi, for some fixed value of iin the range 1 ≤i≤n,
one can form a product of all the distances
∏n
j=1
dj=d 1 d 2 d 3 ···di− 1 didi+1 ···dnand then
remove the term difrom this product to form the weight wi=d 1 d 2 d 3 ···di− 1 di+1 ···dn.
A shorthand notation to represent the above weights is given by the product formula
wi=
∏n
jj=1=i
dj where dj=
√
(x−xj)^2 + (y−yj)^2 + (z−zj)^2 (9 .4)
for j= 1, 2 , 3 ,.. ., n. The vector F at the interpolation position (x, y, z)is then approx-
imated by the weighted average
F=F(x, y, z) = w^1
F 1 +w 2 F 2 +··· +wnFn
w 1 +w 2 +···+wn (9 .5)
Observe that if (x, y, z ) = (xi, yi, zi) for some fixed value of iin the range 1 ≤i≤n,
then di= 0 and equation (9.5) reduces to the identity F(xi, yi, zi) = Fi.The Kriging
method examines the distances between the coordinates of the known quantities and
the selected interpolation point (x, y, z ). It then forms weights where points closest
to the interpolation point have the highest weight. This can be seen by writing the
coefficients of the vectors in equation (9.5) in the form
Coefficienti=
1
di
∑n
j=1
1
dj
(9 .6)
so that the smaller the di= 0, the higher the weighting coefficient. If di= 0, then
all the coefficients with index different from iare zero so that an identity with the
