ETL 1110-1-175
30 Jun 97
b. Measured locations are shown in Fig-
d. The residuals are symmetrically distribu-
ure 5-7 and the basic univariate statistics are listed
ted, (Figure 5-8d). However, the scatterplot (Fig-
in Table 4-1 (water quality A). An initial review
ure 5-8c) indicates that small concentrations are
of the data indicated three important features.
overestimated and that large concentrations are
underestimated. This discrepancy in the estimates
(1) The data seemed to have strong anisotropy
does not indicate an error in the model, but rather,
at about 150 counterclockwise degrees to the east-
indicates a consequence of data that have a large
west baseline.
nugget compared to the sill; in this example the
nugget is approximately 30 percent of the sill. The
(2) The data required a natural log transfor-
large nugget decreases the predictive capacity of
mation so the distribution was approximated by a
the model and increases the smoothing introduced
normal distribution.
by kriging.
e. The established variogram then was used,
(3) No trends were indicated during prelimi-
nary exploration, and ordinary kriging was tenta-
along with the measured locations, to produce
tively selected as the appropriate technique.
ordinary kriging estimates for all points in a 40-by-
20 grid using a grid spacing of about 91-by-91 m.
c. Natural log transformations are routinely
For the kriging procedure, a search radius of about
needed for concentration data that vary over sev-
1,524 m with maximum of 16 and a minimum of
8 locations was specified. Gray- scale maps of
areas of contaminant plumes. The data were
kriging estimates, back transformed to concentra-
transformed to log space and fit acceptable criteria
tions and in log space, as well as the kriging stan-
for normality. After transformation to log space,
dard deviations in log space, are shown in Fig-
the techniques described in Chapter 4 were used
ures 5-9a, 5-9b, and 5-9c.
to guide the following steps for variogram
f. The back-transformation procedure was a
construction:
simple exponentiation of the log space kriging
(1) An exponential model was used to fit a
estimates. Such a back-transformation does not
directional variogram at an angle of 150 counter-
use bias-correction factors to deal with moment
clockwise degrees to the east-west baseline. The
bias and, consequently, the back-transformed
variogram had a nugget of 1.00 log concentration
values must be interpreted as a median value
squared, a sill of 3.20 log concentration squared,
rather than a mean value. The simple back-
and a range of 1,295 m [Figure 5-8a and Table 5-1
transformation, however, is convenient and was
(water quality A)].
performed, principally, to enhance visual inter-
pretation of the kriging estimates. Univariate sta-
(2) An exponential model was also fit to a
tistics for the log-space kriging estimates are listed
directional variogram at an angle of 240 counter-
in Table 5-2 (water quality A). The kriging results
clockwise degrees to the east-west baseline. The
do have noticeable smoothing; however, they also
variogram had a nugget of 1.00 log concentration
indicate a plume emanating from an area just
squared, a sill of 3.20 log concentration squared,
northwest of the center of the area and movement,
and a range of 229 m [Figure 5-8b and Table 5-1
(water quality A)].
estimates are a very good representation of the
results from many other more elaborate studies.
(3) Cross-validation was performed using the
g. An additional comment concerning log
geometric anisotropy of the two variograms and
the results [Figures 5-8c and 5-8d, and Table 5-1
transformations is appropriate. To indicate the
(water quality A)] were acceptable.
effect of the log transform on probabilities in
5-23