is an estimate of the conditional probability distri-

bution *P *[*Z *(*x*0) # *c ** *Z *(*x*1) *Z *(*x*2), ..., *Z *(*x*n)] .

(1) There may be situations when a transfor-

This analysis may be performed for a range of

mation that makes *Z(x) *approximately normal

values of *c*, and by doing this the entire distribution

cannot be easily determined. In such situations,

function can be estimated. This estimate of the

distribution function can be used in the same man-

ner discussed above to obtain prediction intervals

about the probability distribution of *Z(x)*. Because

or estimates of quantiles. For example, to estimate

no distributional assumptions are made, this tech-

nique is known as a **nonparametric **statistical

the value that has a 1-percent chance of being

exceeded at location *x*0, the value of *c *for which the

procedure. An example of indicator kriging is

kriged indicator prediction is 0.99 at that location

included in Chapter 5, and a paper by Journel

is determined.

(1988) is a good reference for additional informa-

tion about indicator kriging.

(4) One advantage of indicator kriging is that

the indicator variogram is robust with respect to

(2) To perform indicator kriging, a special

extreme outliers in the data because no matter how

transformation, known as an indicator transforma-

large (or small) *Z(x) *is, the indicator variable is

tion, is applied to *Z(x)*:

either 0 or 1. Indicator variables may also be used

in the context of block kriging. For example, a

1, *Z*(*x*) # *c*

(2-51)

spatial average of *I(x,c) *over a block *B *equals the

0, *Z*(*x*) > *c*

fraction of block *B *for which *Z(x) *is less than *c*.

Another advantage of indicator kriging is that it

If, as in the usual kriging scenario, the data set at

can be used when some data are censored.

hand consists of measurements of the regionalized

variable *Z(x) *at *n *locations, *c *needs to be fixed

(5) Despite the relative ease of implementa-

first, and then the indicator transformation is

tion, there are several drawbacks to indicator

applied by replacing values that are less than or

kriging, and investigators may wish to use this

equal to *c *with 1 and values that are greater than *c*

technique only when other methods, such as

with 0. The variogram and kriging analysis is then

normality transformations, produce unacceptable

performed using these 0's and 1's rather than the

results. For example, the kriged values of *I(x,c*)

raw data.

may be less than 0 or larger than 1. Also, the

kriged prediction for *I*(*x*,*c*1) may be larger than the

(3) Kriging predictors using the indicator data

kriged prediction for *I*(*x*,*c*2) even if *c*1<*c*2, which is

will be equal to their observed values of 0 or 1 at

not compatible with a valid probability distribu-

the measurement locations *x*i, *i*=1,...,*n*. However,

tion. There are several more advanced techniques

at locations different from the measurement loca-

for dealing with these problems (Chapter 18,

tions, predictions may be between 0 and 1. In

Isaaks and Srivastava (1989); however, they are

interpreting these values, the power of indicator

beyond the scope of this ETL.

kriging becomes apparent. A predicted value at *x*0

2-17

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