at a location *x*0 using information at measurement

(2) To illustrate quantile estimation, suppose

locations *x*1, *x*2, ..., *x*n, so, in terms of conditional

that contaminant concentrations are being studied

distributions, interest focuses on *P *[*Z *(*x*0) # *c ** *Z*

and the concentration that has only a 1-percent

(*x*1), *Z *(*x*2), ..., *Z*(*x*n) ]. The vertical bar denotes the

chance of being exceeded at location *x*0 needs to be

conditioning and is read "given." This conditional

determined. The appropriate (one-sided) value

probability distribution needs to be determined to

from a normal table is 2.33, so the desired estimate

^

is *Z *(*x*0) % 2.33FK (*x*0).

make probability statements about the regionalized

variable at location *x*0. Also, **conditional mean**

(3) Even if *Z(x) *is not Gaussian, it is often

and **conditional variance **can be defined in the

possible to find a **transformation**, *Y(x)=T(Z(x)),*

present context in the same way that mean and

such that *Y*(*x*) is approximately Gaussian. When a

variance for distribution functions were defined in

section 2-2.

transformation is made, the kriging analysis is per-

formed using the transformed data *Y*(*x*), and the

(3) Section 2-6*b *contains methods for using

inverse transformation may be applied to obtain

kriging output to obtain prediction intervals or

prediction intervals for the original data. For

quantiles when the regionalized random variable is

example, the most common transformation is the

(natural) **logarithmic transformation**, in which

either normally distributed or can be transformed

to a near-normal distribution. Section 2-6*c *dis-

for *Z(x) *is then {exp [*^ *(*x*0) - 1.96 Fk(*x*0)], exp [*^*

cusses indicator kriging, which is a nonparametric

method for obtaining quantiles when data cannot

be transformed adequately to a normal distribution.

is a one-to-one function such as a logarithmic

transform, prediction intervals for the original data

can be obtained by simply back-transforming pre-

diction intervals for the transformed data.

(1) For prediction at a location *x*0, a kriging

^

analysis produces the predictor *Z *(*x*0) and the asso-

(4) Although it is a simple matter to obtain

2

ciated kriging variance Fk (*x*0) . If more informa-

prediction intervals and probabilities using simple

back-transformation, it is more difficult to obtain a

tive probability assessments are to be made, the

ideal situation is when *Z(x*) can be assumed to be a

predictor of the untransformed data that is both

unbiased and optimal in some sense. For example,

Gaussian, or normal, process, which means that

in the case of a logarithmic transformation, a

tribution for any set of *n *locations and any value

kriging analysis using the transformed data yields

a predictor *^ *(*x*0), which is the best linear unbi-

of *n*. In this case, the conditional probability dis-

ased predictor of *Y *(*x*0). However, the back-

tribution of *Z*(*x*0) given the *n *observations is a nor-

transformed value ^ (*x*0) = exp [Y (*x*0)] does not

^

Z

mal distribution with conditional mean equal to the

^

kriging predictor *Z (x*0) and conditional variance

possess these same optimality properties as a pre-

2

equal to the kriging variance Fk (*x*0) . This normal

dictor of *Yx*0. The methodology known as log-

distribution can be used to obtain a **prediction**

normal kriging, and more generally trans- normal

kriging, has been developed to obtain predictors in

this setting (Journel and Huijbregts 1978), but

data). For example, from a table of the normal

because of the complexity involved in these pro-

distribution, a value of 1.96 corresponding to a

cedures, they are not usually used by practitioners.

0.95 (two-sided) probability can be obtained.

If a predicted value corresponding to *Z*(*x*0) needs to

Then the assertion that there is a 95-percent chance

that *Z*(*x*0) will be in the 95-percent prediction inter-

be obtained for purposes of contour plotting, the

val [*Z *(*x*0) & 1.96 FK (*x*0), *Z *(*x*0) % 1.96 FK (*x*0)]

kriging predictions *Y *(*x*0) may be back-transformed

^

^

and plotted, as long as the investigator realizes that

can be made. Knowing this interval is much more

such values do not have the usual kriging opti-

useful than simply knowing the kriging predictor

mality properties.

and variance.

2-16

Integrated Publishing, Inc. |