and a fluctuation term Y(x) of mean zero with a
m(x) called the drift,
variogram y(h) called the underlying variogram.
The drift function m(x)
varies slowly and can be modeled, at least locally, as
k
m(x) = > by f°
L
(x)
>
2=0
where the £" (+) stand for known basic functions--monomials in practice-and the b, are unknown coefficients.
Under this model
E[Z(v)} =D, b, ef £*(x)dx = Dorf)
L
Vv
and
E[z (v)] = d BIZ (x,)] =
g
2 dy 2, yf (x,)> so that
&
&
£1.
Gp)
yf
DoylD
=
2@)1
BIz*(v)
g
i
If we want
the bias to be zero whatever the true unknown by
(universal
unbiasedness), we have to impose the conditions
ya,
(x,i ) = £*(v),
~ i
2 = 0,...,k
1
Minimizing E[ >»
A, 20x) - Z(v) ]2 subject to the unbiasedness conditions
i
leads to the Universal Kriging system with k + 1 lagrange parameters Up?
2, yx, - x) +L upto) = (x, , v),
J
Q
2
DAf (x,) = £°),
2 = 0,...,k
i=1,...,N
UNIVERSAL KRIGING
SYSTEM
(9)
L
Naturally, Equation 8 is a special case of Equation 9, provided £9 (x) 21.
At its minimum, the MSE (kriging variance) is
oe = EZ(v) ~ 2(v)]? = Ey Wy 9 + Df)
and provides a measure of the error of estimation.
The inference of the variogram y(h) in the presence of a drift poses a
serious problem.
Indeed, the "raw' variogram
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