11 Ordinary Kriging
Ordinary kriging is the most widely used kriging method. It serves to estimate
a value at a point of a region for whieh a variogram is known, using data in the
neighborhood of the estimation loeation. Ordinary kriging ean also be used to
estimate a block value. With loeal seeond-order stationarity, ordinary kriging
implieitly evaluates the mean in a moving neighborhood. To see this, first a
kriging estimate of the loeal mean is set up, then a simple kriging estimator
using this kriged mean is examined.
Ordinary kriging problem
We wish to estimate a value at Xo as represented on Figure 11.1 using the data
Figure 11.1: A domain with irregularly spaeed sampie points (blaek dots) and a
Ioeation of interest xo.
values from the n neighboring sampie points X o' and eombining them linearly
with weights WO'
n
Z*(xo) = L WO' Z(x O' )
0'=1
Obviously we have to eonstrain the weights to sum up to one beeause in the
extreme ease when all data values are equal to a eonstant, the estimated value
should also be equal to this eonstant.
We ass urne that the data are part of a realization of an intrinsie random
function with a variogram ,(h).
H. Wackernagel, Multivariate Geostatistics
© Springer-Verlag Berlin Heidelberg 1995
Ordinary Kriging 75
The unbiasedness is warranted with unit sum weights
E[ Z*(xo) - Z(xo)] = E[ t
",=1
w'" Z(x",) - Z(xo)· t W"']
",=1
'-..--'
1
n
LW", E[ Z(x",) - Z(xo)]
",=1
= 0
because the expectations of the increments are zero.
The estimation variance O"fu = var( Z*(xo) - Z(xo)) is the variance of the linear
combination
n
Z*(xo) - Z(xo) = LW", Z(x",) - 1 . Z(xo) = LW", Z(x",)
",=1 ",=0
with a weight Wo equal to -1 and
LW",=O
",=0
Thus the condition that the weights numbered from 1 to n sum up to one also
implies that the use of the variogram is authorized in the computation of the
variance of the estimation error.
COMMENT 11.1 The variogram is authorized for ordinary kriging, but not for
simple kriging, because the latter does not include a constraint on the weights.
r]
The estimation variance is
O"~ = E [ (Z*(xo) - Z(xo)
n n n
-i(xo - xo) - L LW", Wß i(X",-Xß) + 2 L W",i(X",-Xo)
"'=1ß=1 ",=1
By minirnizing the estimation variance with the constraint on the weights,
we obtain the ordinary kriging system (OK)
( i(Xn-X1)
i(X1~X1)
i(X1-Xn) ~) (W[) (i(X1~XO))
=
i(Xn-Xn) 1 wn i(Xn-Xo)
1 1 0 /lOK 1
where the w~K are weights to be assigned to the data values and where fLOK is the
Lagrange parameter. The left hand side of the system describes the dissimilarities
between the data points, while the right hand shows the dissimilarities between
each data point and the estimation point xo.
