ETL 1110-1-175
30 Jun 97
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 2-4b and
example, an estimate of the average concentration
2-4c that can be used to directly compute the krig-
of a contaminant over an entire aquifer based on
ing estimate of ZB, 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 ZB 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 (x0 i)
to be smaller than the kriging variance at a single
m
1
(2-48)
ZB =
location.
m i=1
where x0i, i=1,...,m, denotes m prediction locations
2-5. Co-kriging
in block B. The object is to predict this average
rather than the regionalized variable at a single
a. Kriging as discussed so far provides a way
location. In many applications, the locations x0i
of predicting values of a regionalized variable Z(x)
might correspond to nodes of a regular grid or
at a location x0 based on measurements of the same
finite- element nodes in a groundwater model.
variable at locations x1, x2, ..., xn. 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(x0). 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 ZB 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 x0
kriging. In either case, the predicted value of ZB
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 (x0 i)
m
(2-49)
1
^
^
ZB =
are to be used as well. Denote the second variable
m i=1
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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