where it is necessary to estimate the average value

kriging variance is not as simple, because the

of *Z *over an estimation block of much larger area

individual kriging estimates are not independent of

than is represented by an individual sample. For

one another. There are simple modifications to the

kriging equations discussed in sections 2-4*b *and

example, an estimate of the average concentration

2-4*c *that can be used to directly compute the krig-

of a contaminant over an entire aquifer based on

ing estimate of *Z*B, along with its kriging variance

point measurements at various locations might be

needed. In other applications, an estimate of the

(Chapter 13, Isaaks and Srivastava (1989)). The

average concentration of soil contaminant in daily

equations are not presented in this ETL. The com-

excavation volumes that are much larger than the

puter packages described in the next section can be

volume of an individual sample may be needed.

used to compute block kriging estimates. In gen-

Let *Z*B be the average value of *Z*(*x*) over a particu-

eral, kriged values of block averages are less

lar block *B*,

variable than kriged values at single locations.

Consequently, the blocked kriging variance tends

j *Z *(*x*0 *i*)

to be smaller than the kriging variance at a single

1

(2-48)

location.

where *x*0*i*, *i*=1,...,*m*, denotes *m *prediction locations

in block *B*. The object is to predict this average

rather than the regionalized variable at a single

location. In many applications, the locations *x*0*i*

of predicting values of a regionalized variable *Z*(*x*)

might correspond to nodes of a regular grid or

at a location *x*0 based on measurements of the same

finite- element nodes in a groundwater model.

variable at locations *x*1, *x*2, ..., *x*n. In some situa-

Results of the block kriging are dependent on *m*

tions, however, there will be available measure-

and on the placement of the prediction locations.

ments not only of *Z(x)*, but also of one or more

Selecting a large number of locations in block *B*,

other variables that can be used to improve predic-

where each location has approximately the same

tions of *Z*(*x*0). The variable *Z(x) *will be called the

representative area, is the best approach (Chap-

primary variable, because it is the one to be pre-

ter 13, Isaaks and Srivastava (1989).

dicted, and the other variables will be called

secondary variables. **Co-kriging **is the technique

(2) The objective of block kriging is to obtain

that allows the use of the information contained in

the best linear unbiased predictor of *Z*B and an

secondary variables in the prediction of a primary

estimate of the block kriging variance based on the

variable. As an example, suppose that *Z(x) *is a

measurements. The model for *Z*(*x*) can be the

regionalized variable representing the hexavalent

constant-mean model (Equation 2-30) assumed for

chromium concentration, a relatively difficult

ordinary kriging or the more general linear regres-

determination, and that hexavalent chromium con-

sion model (Equation 2-43) assumed for universal

centration needs to be predicted at a location *x*0

kriging. In either case, the predicted value of *Z*B

based on measurements of hexavalent chromium at

coincides with the average of the predicted values

other locations, but there are also measurements of

of the individual measurements in the block; that is

a second relatively easily determined contaminant,

for example lead, that tend to be correlated with

hexavalent chromium concentration and these data

j *Z *(*x*0 *i*)

(2-49)

1

^

^

are to be used as well. Denote the second variable

lead by a regionalized variable *W(x)*, and assume

that measurements have been made on *W *at *m*

In this equation, the individual predicted values are

locations *x*'1 x'2, ..., *x'*m. The co-kriging predictor

obtained from either the ordinary or universal krig-

of *Z(x)*0 is then

ing equations. However, computation of the block

2-14

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