ETL 1110-1-175
30 Jun 97
ZC (x0) = j wi Z (xi)
n
universal kriging, the independent variables in
^
Equation 2-43 need to be known with certainty at
i=1
the prediction location x0. For example, aquifer
(2-50)
thickness might be an independent variable in
% j w 'j W (x'j)
m
predicting aquifer head if it can easily be
j=1
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 wi and w'j so that
the resulting predictor is the best linear unbiased
predictor. Also, as with kriging, co-kriging
2-6. Using Kriging to Assess Risk
requires modeling of the variogram for Z, but
co-kriging presents the investigator with the addi-
a. General.
tional necessity of modeling the variogram of W
and the cross variogram for Z and W. The opti-
(1) The kriging predictor of Z(x0) 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(x0), 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(x0) is concentration of a contam-
inant, the investigator might like to be 95 percent
b. One situation in which co-kriging might be
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
tional probability distribution function of the
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-
c. Although co-kriging is similar to universal
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
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