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 4-9*c*, 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-

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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