ETL 1110-1-175
30 Jun 97
provides a measure of uncertainty for the predic-
(a) In indicator kriging, analysis is performed
tion. In some cases, it may be desirable to go even
using what are known as indicator variables rather
further in specifying the nature of the uncertainty
than the measured data themselves. An indicator
than simply giving the variance. One way to pro-
variable is thus a special kind of transform of the
ceed is to try to obtain what is known as a predic-
measured data and can have only two possible
tion interval. Here one seeks an interval such that
values: 0 or 1. To obtain the indicator variables
there is a certain probability, typically 95 percent,
to be analyzed, first specify a threshold value, say
c, which may represent, for example, a contami-
that the actual value lies in this interval.
nant concentration level which is of particular
(b) Finding such an interval often hinges on
importance. At each measurement location, the
having knowledge of the probability distribution of
indicator variable is then assigned a value of 1 if
the measured value is less than or equal to c, and is
the variables being sampled. One ideal situation is
when the variable of interest, e.g., contaminant
assigned a value of 0 if the measured value is
greater than c. This kind of transform will allow
concentration, can be assumed to have a normal
distribution. In this case, given the set of measured
censored data, or data reported as less than some
values, a potential value at an unsampled location
reporting limit, to be included in the analysis if the
has a normal distribution with mean given by the
reporting limit is less than or equal to the cutoff
value of c. After the indicator transform has been
kriging estimate and variance given by the kriging
variance. It is thus, using classical statistics,
performed, the kriging analysis is performed using
straightforward to use this normal distribution to
these indicator variables in the same manner dis-
obtain a 95 percent prediction interval for concen-
cussed above; first a variogram is obtained, and
tration at the unsampled location.
the kriging equations yield the optimal linear pre-
dictor and the kriging variance for the indicators.
(7) Transformations. Having a prediction
interval will generally be much more informative
(b) Whereas the indicator kriging analysis is
than simply having the kriging estimate and kriging
done using only 0's and 1's, the interpolated esti-
variance, which explains why investigators often
mates are not restricted to these two values. In
ask whether normality assumptions can be made
most cases the estimates are between 0 and 1,
for their data. When a normality assumption
which is interpreted to be the probability that the
actual value is less than or equal to the threshold c.
cannot be made, it is sometimes possible to find a
transformation that will make the data normal, or
Performing this analysis for a number of different
threshold values, c, can give the investigator infor-
nearly so. For example, a transformation that is
often tried is the logarithmic transformation. That
mation about the probability distribution of con-
is, one simply takes the logarithm of all data values
taminant values at a location, which may in turn be
(assuming they are > 0) and performs the geosta-
used to obtain prediction intervals. As discussed
tistical analysis on these transformed values rather
above, such intervals may even be more valuable
than on the original data. Prediction intervals
than having only the optimal predictor and vari-
obtained using transformed values can be readily
ance provided by the usual kriging analysis, partic-
converted to corresponding intervals on untrans-
ularly if behavior of extremes may be of interest to
formed variables. There are, however, subtleties
the investigator. The advantage of using indicator
that must be considered in back-transforming the
kriging to obtain prediction intervals is that it is
kriging estimate and the kriging variance; these are
not necessary to assume a distribution for the data,
discussed in more detail in Chapter 2.
as in the discussion of normality above.
(8) Indicator kriging.
1-8
Previous Page
