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where C is the stationary covariance matrix calculated on the observed values having
entries C ij = cov(x i , x j ) for all i, j = 1, . . . , n. c is the vector of the stationary
covariance between the predicted value and the observed points having entries c i =
cov(x ∗ , x i ). They are determined by the variogram. 1 is the vector of ones.
The stationary covariance function is usually unknown and it is estimated by
defining a variogram such that,
cov(x i , x j ) = C(h) := σ − γ (h),
(12.26)
where h = ||x i − x j || is the spatial distance of x i and x j . γ (h) is the variogram
and σ is the so-called sill which represents the maximum (global) variance of the
observation data.
By substituting Eq. (12.25) into Eq. (12.23) the interpolation surface is
ˆ
f (x) = ˆ
μ + c
T C
−1 (f − ˆ
μ1),
(12.27)
where the mean is
ˆ
μ =
1 T C
−1 f
1 T C
−1 1
,
(12.28)
and the squared standard deviation of the RSM is
ˆ
σ
2
=
(f − ˆ
μ1) T C
−1 (f − ˆ
μ1)
n
.
(12.29)
12.3.3.1 Variogram
The variogram models the variance of random data at the given separation vector h,
as it was defined by [11].
The empirical variance of a data set can be calculated:
s
2
=
1
2
1
n(n − 1)
i =j
(y j − y i )
2 .
(12.30)
The following experimental variogram is obtained by substituting y i = y(x i ) and
y j = y(x i + h) in the above equation:
γ (h) =
1
2
1
N(h)
N(h)
i=1
(y j − y i )
2 ,
(12.31)