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

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

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