ETL 1110-1-175
30 Jun 97
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
d. In sequential Gaussian simulation a set of
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 (x0) - 1.96FK (x0) to 8 (x0) + 1.96FK
Z
Z
path. At each location, a specified set of neighbor-
(x0) discussed in section 2-6b. 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 2-6b. 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
g. A central point that needs to be empha-
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-
e. A sequential algorithm like this may also
sider the problem of determining placement of
be applied in the context of indicator kriging (see
groundwater monitoring wells to detect and moni-
section 2-6c). 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
f. To get an idea of how simulation results
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 x0 (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 x0, which,
7-6
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