variogram. For example, the lag associated with

the maximum of 8 of the residuals can be a good

(

first approximation for the range of the theoretical

variogram.

(1) Parameters of the theoretical variogram

obtained from the initial fitting and refinement of

the sample variogram are calibrated using a krig-

ing cross-validation technique. In this procedure,

adverse effect on 8 . However, divergent data

(

the fitted theoretical variogram is used in a kriging

values can be screened for evaluation using a

analysis in which data values are individually sup-

Hawkins statistic (Hawkins 1980), which is

pressed and estimates made at the location using

described in the context of kriging by Krige and

subsets of the remaining points. As described in

Magri (1982). A neighborhood containing 4 to 10

section 4-3, these subsets are the data points in a

data points, approximately normally distributed,

moving neighborhood surrounding the point under

around each suspected outlier must be defined.

consideration. The calibration estimate made at

Despite potential outliers in the data set, a best

each data location requires a matrix inversion,

guess initial theoretical variogram also is needed.

which could be very time-consuming if all remain-

ing data locations were used to construct the

matrices rather than just those within a neighbor-

paring a suspect datum to the mean value of the 4

hood of a limited search radius.

to 10 surrounding data, the smaller number being

sufficient if the variability is lower. The spacing

(2) After kriged values at all data locations

between these surrounding points is accounted for

have been estimated in the above manner, the data

by the properties of the chosen variogram. A value

are used with their kriged values and kriging stan-

for the statistic of 3.84 or higher would indicate an

dard deviation to obtain cross-validation statistics.

outlier on the basis of a 95-percent confidence

A successful calibration is based on criteria for

interval. A larger number of surrounding points

these statistics, which are described in the next

has the direct effect of increasing the magnitude of

section. If the criteria cannot be reasonably met by

the statistic. Anomalous points are removed from

adjusting the parameters in the given theoretical

the data set and the procedures described for

variogram function, then calibration should be

obtaining the sample variogram are repeated for

reinitialized with a different theoretical variogram

the smaller data set. There were no outlier prob-

function. In some data sets with nonstationary

lems in the Saratoga data.

spatial means, the drift polynomial may have to be

changed as well as the variogram to achieve a

satisfactory calibration.

regarding the merit of automated outlier-detection

methods. A procedure such as that described here

is presented as an investigative tool with the under-

(1) The kriging cross-validation error *e*i cor-

standing that the investigator will also use atten-

responding to measurement *z(x*i) is defined as

dant justification along with a Hawkins-type

statistic to ultimately decide if a data value is

(4-3)

discarded as a true outlier or retained as a valid

observation. In some situations, highly problem-

atic data values are removed for computation of

where ^ (*x*i) is the kriged estimate of *z *(*x*i) based

the sample variogram points but are reinstated for

on the remaining *n*-1 measurements in the data set.

kriging.

4-14

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