ETL 1110-1-175
30 Jun 97
at a location x0 using information at measurement
(2) To illustrate quantile estimation, suppose
locations x1, x2, ..., xn, so, in terms of conditional
that contaminant concentrations are being studied
distributions, interest focuses on P [Z (x0) # c * Z
and the concentration that has only a 1-percent
(x1), Z (x2), ..., Z(xn) ]. The vertical bar denotes the
chance of being exceeded at location x0 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 (x0) % 2.33FK (x0).
make probability statements about the regionalized
variable at location x0. 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 2-6b 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 2-6c dis-
Y(x)=1n[Z(x)]. A 95-percent prediction interval
for Z(x) is then {exp [^ (x0) - 1.96 Fk(x0)], exp [^
Y
Y
cusses indicator kriging, which is a nonparametric
(x0) + 1.96 Fk(x0)]}. As long as the transformation
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
b. Normal distributions and transformations.
can be obtained by simply back-transforming pre-
diction intervals for the transformed data.
(1) For prediction at a location x0, a kriging
^
analysis produces the predictor Z (x0) and the asso-
(4) Although it is a simple matter to obtain
2
ciated kriging variance Fk (x0) . 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 ^ (x0), which is the best linear unbi-
Y
of n. In this case, the conditional probability dis-
ased predictor of Y (x0). However, the back-
tribution of Z(x0) given the n observations is a nor-
transformed value ^ (x0) = exp [Y (x0)] does not
^
Z
mal distribution with conditional mean equal to the
^
kriging predictor Z (x0) and conditional variance
possess these same optimality properties as a pre-
2
equal to the kriging variance Fk (x0) . This normal
dictor of Yx0. The methodology known as log-
distribution can be used to obtain a prediction
normal kriging, and more generally trans- normal
interval for Z(x0) (conditional on the measured
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(x0) needs to
Then the assertion that there is a 95-percent chance
that Z(x0) will be in the 95-percent prediction inter-
be obtained for purposes of contour plotting, the
^
val [Z (x0) & 1.96 FK (x0), Z (x0) % 1.96 FK (x0)]
kriging predictions Y (x0) 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
Previous Page
