(1) Suppose the extent of groundwater con-

tamination of a particular pollutant over a given

study area is being determined. To simplify the

presentation, all data are assumed to be distributed

the necessary theoretical background for under-

over a two-dimensional region. In three-

standing kriging applications. Emphasis will be

dimensional groundwater flow systems, one could

placed on presentation of the basic ideas; long

study the depth-averaged concentration of a pol-

formulas or derivations are kept to a minimum.

lutant or the concentration of the pollutant in a par-

Statistical terms that are commonly used in

ticular horizontal stratum of the flow system. Let

geostatistical applications will be highlighted with

a vector *x=(u,v) *denote an arbitrary spatial loca-

bold text and briefly defined as they are intro-

tion in the study area. Unless otherwise stated, it

duced; notation used in this ETL is also tabulated

will be assumed throughout the ETL that *u *is the

in Appendix B. The reader who wishes a more

east-west coordinate and *v *is the north-south

thorough discussion of these fundamental concepts

coordinate (Figure 2-1). Denote by *z(x) *a meas-

may consult the references cited in Chapter 3.

urement at location *x*, such as the concentration of

Previous exposure to engineering statistics at the

a pollutant. The ultimate goal of an investigator

level of Devore (1987) and Ross (1987) would be

would be to determine *z(x) *for all locations in the

helpful in understanding some parts of this chap-

study area. However, without explicit knowledge

ter. Readers with limited statistical experience

of the flow and transport field, this goal cannot be

may wish to briefly scan this chapter and refer

achieved. Therefore, suppose, instead, that the

back to it after reading the remaining chapters.

goal is to estimate the values of *z(x) *with a given

error tolerance. In other situations, small estima-

tion error over some parts of the study area (for

ables are discussed. Regionalized random varia-

instance, near a domestic water supply) may need

bles constitute the random process that is sampled

to be obtained, while allowing larger estimation

to obtain the observed data that are available for

errors in other parts of the study area. The theory

analysis. Basic ideas related to probability distri-

of regionalized random variables is designed to

butions, means, variances, and correlation are

accomplish these goals.

introduced. The variogram, which is the funda-

mental tool used in geostatistics to analyze spatial

(2) In the regionalized random variable theory,

correlation, is introduced in section 2-3. In sec-

the true measurement *z(x) *is assumed to be the

tion 2-4 how kriging is used to obtain the best

value of a **random variable ***Z(x)*. Associating a

weights for spatial prediction is discussed, and

random variable *Z(x) *with a true measurement *z(x*)

how the mean squared prediction error for these

is done for the purpose of characterizing the degree

predictions is computed is also shown. Section 2-5

of uncertainty in the quantity of interest at point *x*.

deals briefly with co-kriging, which is prediction of

If there is no actual measurement taken at *x*, then

one variable based not only on measurements of

the values taken on by *Z(x) *represent "potential"

that variable but on measurements of other vari-

measurements at *x*; that is, *Z(x) *represents possible

ables as well. Finally, section 2-6 shows how

values that might be expected if a measurement

kriging may be applied to determine not just opti-

were taken at *x*. Because there is uncertainty asso-

mal spatial predictions but also probabilities

ciated with *Z(x)*, it needs to be characterized by a

associated with various events, such as extreme

events that may be of importance in risk-based

where *P *denotes probability and *c *is any constant.

analyses.

2-1