The kriged estimate is obtained by ordinary kriging

results is the unbiasedness condition where

j *e*i . 0.

if the spatial mean is constant or by universal krig-

1

ing if the spatial mean is not stationary. A reason-

able criterion for selecting a theoretical variogram

would be to minimize the squared errors, j *e*i ,

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, *e*i

ances, Fk (*x*i) , where Fk (*x*i) is the kriging vari-

=1,2...,*n*, are approximately normally distributed

ance at location *x*i. 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

dard normal, outliers not detected using the method

2

(4-4)

=

~

discussed in section 4-7 may be detected by com-

FK

(*x *)

paring the absolute values of the reduced kriging

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 *e*i

mated measurement error related to obtaining

1

2

min

water levels at the well depths in the Saratoga

(4-5)

valley.

0.5

j *e*i

1

~2

(5) After two iterations using drift residuals,

.1

subject to

as described in section 4-7, a final variogram was

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

root-mean-squared error of 0.5794. The

1978). An additional check on the cross-validation

4-15