ETL 1110-1-175
30 Jun 97
c. Indicator kriging.
is an estimate of the conditional probability distri-
bution P [Z (x0) # c * Z (x1) Z (x2), ..., Z (xn)] .
(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
indicator kriging can be used to make inferences
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 x0, 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
I (x,c) =
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,c1) may be larger than the
(3) Kriging predictors using the indicator data
kriged prediction for I(x,c2) even if c1<c2, which is
will be equal to their observed values of 0 or 1 at
not compatible with a valid probability distribu-
the measurement locations xi, 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 x0
2-17
Previous Page
