ETL 1110-1-175
30 Jun 97
reduced-root-mean-squared error is too small,
parameters, generally, also will have an effect on
indicating that the kriging variances produced by
the mean-squared error. The larger the nugget is
the model are too large compared to the actual
as a percentage of the sill, the larger the mean-
squared errors. This fact, coupled with the rather
squared error will be. In general, improvements in
large root-mean-squared error, makes the theo-
one statistic are usually made at the expense of the
retical variogram model unacceptable. In sec-
other statistics. The optimization of the statistics
tion 4-9c, a Gaussian variogram is fitted to the
as a set is, in effect, a trial and error procedure that
data that produces much better cross-validation
is operationally convergent.
results than the results for the spherical variogram.
(4) Reduced kriging errors may not approxi-
c. Variogram-parameter adjustments.
mate a standard normal distribution. If this is the
case, a transformation of the data may be needed
(1) If any of the cross-validation statistics
to achieve a more normal distribution, and the
vary unacceptably from their suggested values,
variogram estimation procedure would be repeated.
minor adjustments to the variogram parameters
can be made to attempt to improve the statistics.
(5) Because no convergence could be reached
Whatever modifications are made to the param-
for parameter values of a spherical variogram for
eters, they should not have to be so severe that the
the Saratoga data, a Gaussian theoretical vario-
variogram function drastically deviates from the
gram was fitted to the sample variogram of
sample variogram points. If the support of the
residuals in Figure 4-4. This choice was made
sample variogram points is compromised in order
because the initial sample variogram points could
to achieve acceptable cross-validation results with
be interpreted to have a slight upward concavity,
the given drift-variogram model, a different drift-
but eventually reached a sill. This behavior can be
variogram combination should be investigated.
attributed to correlation rather than to further drift.
After an iterated cross-validation with the Gaus-
(2) A reduced root-mean-squared error that is
sian parameters, a Gaussian variogram with a
nugget of 1.49 m2, a sill of 185.81 m2, and a range
unacceptable may be improved upon by adjusting
the range parameter or the nugget value of the
of 27.52 km (Figure 4-8) yielded a root-mean-
variogram. Modifying the range parameter should
squared error of 2.33 m and a reduced-root-mean-
be considered first and any shifts in the nugget
squared error of 1.083. The mean cross-validation
value should be minimal and made only as a final
error is 0.0195 m. These values represent an
recourse. Calibration errors are relatively insen-
improvement over the spherical variogram and
sitive to minor adjustments of the sill.
were deemed acceptable for the Gaussian
variogram.
(3) If the reduced root-mean-squared error is
too small, as in the Saratoga example, extending
(6) A probability plot of the reduced kriging
the range (equivalent to decreasing the slope for a
errors using the final Gaussian variogram is shown
linear variogram) will decrease the kriging vari-
in Figure 4-9. It is reasonably linear between two
ance and thus increase the reduced root-mean-
standard deviations and, thus, approximates a
squared error. If a shift in the nugget value is
standard-normal-distribution function. Finally, a
required, a decrease in the nugget will reduce the
plot in Figure 4-10 of the data versus their kriged
kriging variance. If the reduced root-mean-
estimates indicates that the linear drift-Gaussian
squared error is too large, then a contraction of the
variogram model selected for the Saratoga data
range or a positive shift in the nugget value can be
would produce accurate estimates of groundwater
made, keeping in mind the above caveat of priority
elevations for interpolation or contour gridding in
and extent of the changes. Changes in these
the region.
4-17
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