turning-bands method (Deutsch and Journel 1992,

when analyzed in histogram form, approximates

Journel and Huijbregts 1978).

the probability distribution of potential measure-

ments at that location. If an interval with exactly

25 (2.5 percent) of the values less than the lower

grid points for which simulated values are desired

end and 25 of the values larger than the upper end

is defined and the points are addressed sequentially

were constructed, the interval would almost corre-

from location to location along a predetermined

spond, as expected, to the 95-percent prediction

interval to *8 *(*x*0) - 1.96FK (*x*0) to *8 *(*x*0) + 1.96FK

path. At each location, a specified set of neighbor-

(*x*0) discussed in section 2-6*b*. Thus, for this single

ing conditioning data is retained, including the

original data and simulated grid-location values at

location, the simulation has not produced much

previously traversed grid locations along the path.

more information than kriging alone would have

Then, a random number is generated from a

produced. The real value of simulation, however,

Gaussian distribution with conditional mean and

is that realizations not just at a single location, but

variance determined using a kriging algorithm, and

at all of the grid locations jointly, are obtained.

the value of the random number determines the

These realizations can be used to calculate proba-

simulated process at this location. The conditional

bilities associated with any number of spatial loca-

Gaussian distribution used in simulation is identi-

tions together. For example, the probability that

cal to the conditional distribution discussed in

the largest (maximum) contaminant value over a

section 2-6*b*. An idea of the computational

certain subregion is greater than a particular con-

requirements can be obtained from the fact that a

centration might be assessed. (If the word "larg-

kriging algorithm needs to be applied for each

est" here were replaced with "average," then block

simulation location. For multiple realizations, if

kriging could be used to obtain the answer.)

the path connecting the grid points is kept the

same, the kriging equations need to be solved for

only the first simulation. However, implementa-

sized is that simulation is especially useful when

tion of this procedure needs to take into considera-

probabilities associated with complicated, usually

tion the assumptions concerning the existence of

nonlinear, functions of the regionalized variables

drift; the details of such an implementation are

over a region need to be analyzed. The maximum

beyond the scope of this ETL.

function mentioned in the preceding paragraph is

one simple example. For another example, con-

sider the problem of determining placement of

be applied in the context of indicator kriging (see

groundwater monitoring wells to detect and moni-

section 2-6*c*). At each grid point along the path, a

tor groundwater contamination emanating from a

(Bernouli) random variable taking on only two

potential point source. Given an existing set of

possible values, 0 or 1, is generated, with the rela-

hydraulic-head data, kriging might be applied and

tive probability of these two values being deter-

flow lines determined from resulting hydraulic-

mined by indicator kriging applied, as in the

head gradients. Intersection of the flow line from

previous paragraph, to the original observed indi-

the point source with the regional boundary then

cator data and the previously simulated indicator

might be used to determine monitoring well place-

values.

ment. Conditional simulation would be useful to

determine uncertainty associated with location of

well placement or to give an indication of how

might be used in a risk-assessment setting, assume

many monitoring wells might be appropriate. In

again that the underlying process is Gaussian and

this case, the variable of interest, well location, is a

that 1,000 conditional realizations have been

complicated function of hydraulic heads so this is a

generated. If a single grid point *x*0 (which is not a

problem for which simulation is well-suited. The

measurement point) is considered, then the simu-

reader may refer to Easley, Borgman, and Weber

ation has produced 1,000 values at *x*0, which,

7-6

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