Because the variogram is the same for all *h*>0 and

(2) Suppose the investigator wants to predict

the value of *Z(x*0) by using a **linear predictor**,

there is no trend in the data, there is no reason to

^

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 2-4*b*, deals with the constant-mean model

(2-31)

^

(assumption in Equation 2-30) in which the vari-

0

ogram is not a pure nugget variogram. The

weights of the best linear unbiased predictor will

where *w*i is the weight assigned to *Z(x*i). 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 ^ (*x*0) to be a good pre-

mean. In section 2-4*c*, universal kriging, which is

dictor of *Z(x*0). The first criterion is that ^ (*x*0) be

the extension of ordinary kriging to the case of a

an **unbiased predictor **of *Z(x*0), 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)

kriging that will be discussed in this section are

indicator kriging (section 2-6*c*), block kriging (sec-

tion 2-4*d*), and co-kriging (section 2-5).

(3) An unbiased predictor will neither consis-

tently overpredict nor underpredict *Z(x*0) 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

(4) The smaller the prediction variance, the

closer Z (*x*0) will be (on average) to the true value

(1) General.

Z(*x*0). The geostatistical method of kriging deals

with computing the **best linear unbiased pre-**

(a) Let *Z(x)*be a regionalized random variable

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

(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*(*x*0) 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 *(*x*i)

tion 2-32 holding. The unbiased condition is

(2-34)

1

^

0

2-10

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