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.

(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.

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

eral orders of magnitude, which is common in

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

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

as well as some dispersion, to the southeast; the

(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

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

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