(6) To summarize, the basic model frame-

will adopt the variogram as the primary tool for

work that will be used throughout the ETL is the

analyzing spatial dependence in the remainder of

following: the value of a measurement *z(x) *(con-

this ETL.

centration, porosity, hydraulic head, and so on) at

location *x *of a two-dimensional region is the value

of a regionalized random variable, *Z(x)*, with mean

tion, it is necessary to distinguish between the

( *x*) and stationary covariance function *C(h,a).*

theoretical variogram, which is a population

Other assumptions may be added in the applica-

parameter, and the sample variogram, which is an

tions sections to analyze specific data sets, but this

estimator of the theoretical variogram obtained

from observed data. The **theoretical variogram**

framework will be the basic framework from

of a regionalized random variable, ((*x*1 ,*x*2) is

which many of the results will be derived. In some

situations, the covariance stationarity assumption

defined as one half of the variance of the difference

between residuals at locations *x*1 and *x*2:

may be relaxed, for instance, when using the linear

variogram described in the next section.

1

((*x*1, *x*2) =

Var [*Z * ( (*x*1) & *Z *((*x*2)]

(2-17)

2

Because the residuals have been mean-centered, as

shown in Equation 2-16, they have a mean of zero.

classical (ordinary least-squares) regression

Therefore, using the well-known formula for the

models in that the **residuals**, defined as the devi-

variance of a random variable *X*

ations of the regionalized random variable from its

mean and denoted by

Var (*X*) = *+ X *2 & (*+X*)2

(2-18)

(2-16)

it is seen that Equation 2-17 is equivalent to

are related to one another, whereas the residuals in

1

( (*x*1, *x*2) =

a regression model are generally assumed to be

2

independent. Thus, in the regionalized random-

variable model, observed values of the residuals

The theoretical variogram is always non-negative,

from sampled locations contain valuable informa-

with a small value of *g *indicating that the residuals

tion when predicting the value of *Z(x) *at unsam-

at locations *x*1 and *x*2 tend to be close and a large

pled sites. The relationship among the residuals

value of 8 indicating that the residuals tend to be

can be understood by examining the variogram,

different. Equation 2-19 is sometimes called a

which is a tool that is widely used in geostatistics

for modeling the degree of spatial dependence in a

, but will be referred to in this ETL as a

regionalized random variable. Although the vario-

variogram.

gram is closely related to the covariance function,

there are some important differences between the

variogram and covariance function that will be

variogram before taking observations, but unfortu-

described below. The covariance function, and

nately, it must be estimated using sample data. To

related correlation function, are more commonly

facilitate variogram estimation, it is usually

used in basic statistics courses than the variogram,

assumed in a similar manner to the covariance

so many readers may be more familiar with the

function that ( depends only on the lag,

former concepts. However, the variogram is more

widely used in geostatistics, and because of this we

2-6