universal kriging, the independent variables in

^

Equation 2-43 need to be known with certainty at

the prediction location *x*0. For example, aquifer

(2-50)

thickness might be an independent variable in

% j *w *'j W (*x*'j)

predicting aquifer head if it can easily be

determined at any location. However, aquifer

thickness may need to be considered a secondary

This is a straightforward extension of the kriging

variable in a co-kriging procedure if it is only

predictor in Equation 2-31. Analogous to kriging,

known at a few selected points in the aquifer.

co-kriging produces the weights *w*i and *w'*j so that

the resulting predictor is the best linear unbiased

predictor. Also, as with kriging, co-kriging

requires modeling of the variogram for *Z*, but

co-kriging presents the investigator with the addi-

tional necessity of modeling the variogram of *W*

and the **cross variogram **for *Z *and *W*. The opti-

(1) The kriging predictor of *Z*(*x*0) has certain

mal weights are then expressed in terms of all

desirable properties with respect to how close it is

these variogram properties. More than one sec-

to the actual value of *Z*(*x*0), it is unbiased and has

ondary variable may be included in the co-kriging

smallest variance among all linear predictors. On

predictor, and theory has been developed for

the average, or in an expected sense, the predicted

co-kriging in the presence of drift (universal

value will be near the actual value. When possi-

co-kriging) and block co-kriging. Details are not

ble, however, the investigator would like to go fur-

included in this ETL, but the interested reader may

ther in specifying the relationship between the

refer to Isaaks and Srivastava (1989) and Deutsch

predicted and observed values. Ideally, the investi-

and Journel (1992) for more discussion and cita-

gator would like to make probability statements.

tion of other references.

For example, if *Z*(*x*0) is concentration of a contam-

inant, the investigator might like to be 95 percent

certain that the true concentration is within

useful is when the primary variable is undersam-

0.05 ug/R of the predicted concentration. In other

pled, so any additional information, such as that

situations, the probability that the actual concen-

given by secondary variables, would be helpful.

tration exceeds a given target value might need to

However, although co-kriging can be a useful tool,

be estimated. Knowledge of the entire distribution

joint modeling of several variables tends to be

function of *Z*(*x*), as opposed to knowledge of only

demanding in terms of data and computational

its mean and variogram, can be used for risk-

requirements. Thus, undersampling of the primary

qualified inferences in situations when extremes

variable may present problems for co-kriging as

might be of more interest than averages.

well as for one-variable kriging. Also, unless the

primary variable of interest is highly correlated

(2) Introduction of the concept of a **condi-**

with the secondary variable(s), the weights

assigned to the secondary variable(s) are often

regionalized variable *Z*(*x*) is appropriate at this

small, with the result that the effort needed to

point. This concept will also be used in Chapter 7

include the additional variable(s) may not be

when conditional simulation is discussed. The

worthwhile. For these reasons, co-kriging tends

conditional probability distribution function has a

not to be used extensively in practice.

definition much like that of the probability distri-

bution function in section 2-2, except the proba-

bility that *Z(x) # c *is computed "conditional on,"

kriging, in that both techniques use extra variables

or "given," information at other spatial locations.

to help predict *Z*(*x*), there is an important

The interest in geostatistics is to make predictions

distinction between the two techniques. In

2-15