ETL 1110-1-175
30 Jun 97
Because the variogram is the same for all h>0 and
(2) Suppose the investigator wants to predict
the value of Z(x0) by using a linear predictor,
there is no trend in the data, there is no reason to
^
Z (x0), which is defined as a weighted linear combi-
favor any of the measurements over any of the
other measurements. Therefore, the weights are all
nation of the measured data
the same. Ordinary kriging, which is discussed in
section 2-4b, deals with the constant-mean model
Z (x ) = j wi Z (x )
n
(2-31)
^
(assumption in Equation 2-30) in which the vari-
i
0
ogram is not a pure nugget variogram. The
i=1
weights of the best linear unbiased predictor will
where wi is the weight assigned to Z(xi). To deter-
reflect the information in the variogram and will
result in an improved predictor over the sample
mine specific values for the weights, some criteria
need to be specified for ^ (x0) to be a good pre-
mean. In section 2-4c, universal kriging, which is
Z
dictor of Z(x0). The first criterion is that ^ (x0) be
Z
the extension of ordinary kriging to the case of a
an unbiased predictor of Z(x0), which is expres-
nonconstant mean, is discussed. Universal kriging
is a very powerful tool that can be used to combine
sed as
regression models and spatial prediction into one
unifying theory. Other, more specialized types of
^
(2-32)
+ [Z (x0) & Z (x0)] = 0
kriging that will be discussed in this section are
indicator kriging (section 2-6c), block kriging (sec-
tion 2-4d), and co-kriging (section 2-5).
(3) An unbiased predictor will neither consis-
tently overpredict nor underpredict Z(x0) because
the statistical expectation of the prediction errors is
(6) Before giving the kriging equations, one
zero. The second criterion for a good predictor is
final note is in order. There is a prediction tech-
that it have small prediction variance, defined by
nique in geostatistics known as simple kriging,
which deals with best linear unbiased prediction in
the case when the mean of Z(x) is fixed and known.
^
Var Z (x ) & Z (x )
Simple kriging is not discussed in this ETL,
0
0
(2-33)
because, in most applications, the mean is not
^
= + Z (x ) & Z (x ) 2
known and has to be estimated.
0
0
b. Ordinary kriging.
(4) The smaller the prediction variance, the
^
closer Z (x0) will be (on average) to the true value
(1) General.
Z(x0). The geostatistical method of kriging deals
with computing the best linear unbiased pre-
(a) Let Z(x)be a regionalized random variable
dictor of Z(x0), which is the linear unbiased pre-
with constant mean (Equation 2-30) and isotropic
dictor (Equations 2-31 and 2-32) with the smallest
variogram (Equation 2-20). Also, assume that the
possible prediction variance (Equation 2-33).
variogram reaches a sill so that the variance of
Z(x) is C(0)=s, and the correlation function is
(5) The form of the best linear unbiased pre-
given by Equation 2-28. Although the prediction
dictor will depend on the mean of Z(x). For exam-
equations can be expressed in terms of the vario-
ple, if Z(x) has a constant mean (Equation 2-30)
gram, they will be defined here in terms of the sill
and a pure nugget variogram [((h)=s for all h>0],
(variance) and the correlation function.
the best linear unbiased predictor of Z(x0) will
simply be the average of the measured data
(b) Consider linear unbiased predictors of the
form of Equation 2-31 with the condition in Equa-
j Z (xi)
n
tion 2-32 holding. The unbiased condition is
(2-34)
1
^
Z (x ) =
0
n i=1
2-10
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