Therefore, when *Z(x) *has a stationary covariance

function, the variance of *Z(x) *is constant for all *x*.

(2-9)

The covariance function can then be standardized

(*u*1 & *u*2)2 % (*v*1 & *v*2)2

by dividing it by the variance. The resulting

dimensionless function of *h *is called the **spatial**

Under this assumption, *C(h) *can be estimated by

pooling all pairs of observations that are approxi-

mately *h *units apart and computing a **sample**

D (*h*) =

(2-14)

^

The correlation function is a scale-independent

measure of linear association between values of *Z*

(2-10)

at different locations. The spatial correlation is

& *Z Z*(*x *) & *Z *:

always between -1 and +1, with a value of zero

indicating no linear association.

(5) In addition to being stationary, the covari-

where *h*ij is the distance between *x*i and *x*j and the

ance function in Equation 2-9 has another import-

average is over all pairs of points such that *h*ij is

ant property. It is also **isotropic**, or **omni-**

between *h*-)*h *and *h*+)*h*. The distance *h *is called

the lag and )*h *is called the **lag tolerance**. There

direction between the two locations. In many

are more effective ways to estimate *C(h) *other than

HTRW applications, the correlation between

values of *Z *at two locations is a function of direc-

using Equation 2-10; for example, see Isaaks and

Srivastava (1989). However, because the empha-

tion as well as lag. For example, contaminant

sis in this ETL is on the variogram (to be defined

concentrations in a groundwater flow system might

below) rather than the covariance function, we will

be more highly correlated along a transect in the

not need to use the estimated covariance function.

direction of flow than along a transect perpen-

dicular to the flow. In that case, the covariance

(4) A covariance function is called **stationary**

function depends on both the lag *h *and the angle *a*

if it does not depend on the origin of the coordinate

between locations,

system, that is,

(2-11)

1

2

1

2

(*u*1 & *u*2)2 % (*v*1 & *v*2)2, (2-15)

for any given vector, *b *(Figure 2-1). The covari-

ance function (Equation 2-9) is stationary because

changing the origin does not change the distance

between the points. Substituting *x*1 = *x*2 = *x *in

Equation 2-9 yields

Here, *a *is the angle measured counterclockwise

from the east direction (Figure 2-1). In many geo-

(2-12)

statistical publications or computer packages, the

angle may be defined as clockwise from the north

which, combined with the definitions in Equa-

direction, so care should be taken in defining the

tions 2-6 and 2-8, becomes

appropriate angle in any application. A covariance

function satisfying Equation 2-15 is called **aniso-**

(2-13)

F2 (*x*) = *C *(0) for all *x*

2-5