converting, or back-transforming, kriging esti-

(3) A spherical model also was fit to an

mates, the kriging estimates and the kriging stan-

anisotropic variogram at an angle of 240 deg

dard deviations, in log space, were used to estimate

counterclockwise to the east-west baseline. The

the one-sided 95th percentile at each kriging-

variogram had a nugget of 0.05 indicator units

estimate location according to the formula:

squared, a sill of 0.25 indicator units squared, and

a range of 213 m [Figure 5-10b and Table 5-1

^

(water quality B].

(5-1)

exp *Z *(*x*0) % 1.645FK (*x*0)

^

indicator transform of the measured data, was used

to produce ordinary kriging estimates for the same

^

where (*Z *x0) ( is the kriging estimate at location *x*0,

grid and search criteria as the first water-quality

in log space, and Fk(*x*0) is the corresponding krig-

example. A gray-scale map of the kriging esti-

mates is shown in Figure 5-11. The kriging indi-

ing standard deviation in log space. The resulting

cator map provides a gridded estimate for the

map is shown in Figure 5-9d. Such a map can be

probability of contaminant values being less than

used to indicate areas where the true concentration

the indicator cutoff, which is a concentration of

has only a 5-percent chance of exceeding the value

270 in this example.

shown.

indicator kriging example is probably higher than

transformation, as described in Chapter 2, was

many investigators involved in HTRW site investi-

applied. An indicator cutoff equal to the median

gations would like to use. In this case the number

value of 270 for the untransformed measured data

of measurements [66 in Table 4-1 (water qual-

was selected. The model for indicator kriging esti-

ity B)] used in this example, which is probably a

mates the probability that the concentration would

high number of measurements for typical HTRW

be less than the indicator cutoff. The techniques

site investigations, would not permit construction

described in Chapter 4 were used to guide the fol-

of an indicator variogram for indicator values

lowing steps in variogram construction:

much lower than the median. An alternative to this

problem would be to assume that the log-

(1) No trends were indicated during prelimi-

transformed kriging model developed in the first

nary exploration, and ordinary kriging was tenta-

water-quality example is correct and to rely on the

tively selected as the appropriate technique.

kriging estimates from that model to determine

areas greater than or less than some indicator

(2) A spherical model was used to fit an

value. The same estimates also could be used to

anisotropic variogram at an angle of 150 deg

compute the probability that the concentration was

counterclockwise to the east-west baseline. The

less than some arbitrarily selected value.

variogram had a nugget of 0.05 indicator units

squared, a sill of 0.25 indicator units squared, and

a range of 610 m [Figure 5-10a and Table 5-1

(water quality B)].

5-33