ETL 1110-1-175
30 Jun 97
The kriged estimate is obtained by ordinary kriging
results is the unbiasedness condition where
j ei . 0.
if the spatial mean is constant or by universal krig-
1
ing if the spatial mean is not stationary. A reason-
n
able criterion for selecting a theoretical variogram
would be to minimize the squared errors, j ei ,
2
(3) As indicated in Chapter 2, if probabilistic
with respect to the variogram parameters. How
statements concerning an actual value of Z at an
ever, unlike ordinary least-squares regression,
unmeasured location are to be made relative to the
which also minimizes the sum of squared errors,
kriged estimate and the kriging variance at the
simply minimizing the squared errors is not suffi-
location, it is necessary to explore the distribution
cient for kriging because the resulting model can
of the cross-validation kriging errors. In particu-
~
yield highly biased estimates of the kriging vari-
lar, it is desirable that the reduced errors, ei
2
2
ances, Fk (xi) , where Fk (xi) is the kriging vari-
=1,2...,n, are approximately normally distributed
ance at location xi. This simple minimization
with mean 0 and variance 1. A histogram or nor-
would give unrealistic measures of the accuracy of
mal probability plot of the reduced kriging errors
the kriging estimates. To guard against such bias,
can be used to assess the validity of assuming a
an expression for the square of a reduced kriging
standard normal distribution for the reduced krig-
error is defined:
ing errors. Additionally, if the distribution of
reduced kriging errors can be assumed to be stan-
2
ei
dard normal, outliers not detected using the method
2
(4-4)
ei
=
~
discussed in section 4-7 may be detected by com-
2
FK
(x )
paring the absolute values of the reduced kriging
i
errors to quantiles of the standard normal
distribution.
where the kriging variances are computed using
either Equation 2-36 or 2-47. If the kriging vari-
(4) Using the Saratoga data, a spherical vario-
ance is an unbiased estimate of the true mean-
gram was fitted to the refined sample variogram of
squared error of estimate, then the reduced kriging
the residuals. The estimated nugget was about
errors would have an average near one. Therefore,
1.49 m2, the sill was 133.8 m2, and the range was
the standard cross-validation procedure for evalu-
about 48 km. Because of difficulty in determining
ating a theoretical variogram is:
an exact extrapolated value for the nugget, the
value of 1.49 m2 was selected based on an esti-
0.5
j ei
n
mated measurement error related to obtaining
1
2
min
water levels at the well depths in the Saratoga
n i =1
(4-5)
valley.
0.5
j ei
n
1
~2
(5) After two iterations using drift residuals,
.1
subject to
as described in section 4-7, a final variogram was
n i =1
chosen with a nugget of 1.49 m2, a sill of 148.6 m2,
and a range of 44.8 km (Figure 4-7). These
(2) The expression to be minimized is called
parameters defined the theoretical variogram used
the kriging root-mean-squared error and the con-
to obtain the cross-validation errors using univer-
straint is called the reduced root-mean-squared
sal kriging with an assumed linear drift. The best
error. The reduced root-mean-squared error
combination of statistics that could be obtained
should be well within the interval having endpoints
after several attempts at refining the model were a
2
2
root-mean-squared error of 3.45 m and a reduced
1% 2
and 1 & 2
(Delhomme
n
n
root-mean-squared error of 0.5794. The
1978). An additional check on the cross-validation
4-15
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