(2) In example 1, no assumptions were needed

(1) If the scenario presented in example 1 is

again used, the sample variance *S*n2(*x*) of the *n*

concerning whether the mean changed with spatial

location, because all sampling was done at one

measurements could be computed as follows:

sampling location *x*. In most HTRW applications,

j

the mean will probably change depending on the

1

2

(*x*) =

sampling location. In addition, usually only one

(2-7)

observation is available at any particular location.

Therefore some assumptions regarding the struc-

2

ture of ( *x*) must be made. For example, it is

sometimes appropriate to assume (*x*) ' is

This number gives a measure of dispersion of the

constant for all *x*, in which case *Z(x*) is said to

have a **stationary mean**. Data which have no

variance depends on *n *and on the particular values

underlying trend such as hydraulic conductivity in

observed for *Z*1(*x*), *Z*2(*x*), ..., *Z*n(*x*). However, in

a homogeneous aquifer, for example, might be

the limit as *n *increases, *S*n2(*x*) gets closer and

assumed to have a constant mean. If the mean is

closer to a constant value, which is denoted by

constant, it makes sense to estimate it with the

F2(*x*). In this case, F2(x) is a population param-

sample average of *n *observations taken at different

eter, and *S*n 2(*x*) is a random variable.

spatial locations *x*1, *x*2, ..., *x*n

(2) The mean and variance can both be calcu-

1

lated from the probability distribution of *Z(x)*.

(2-5)

Again, in geostatistics, the relations among region-

alized variables at different locations are of

% ... % *Z *(*x * n)

interest. From the joint distribution of *Z*(*x*1) and

However, in contrast to example 1, *Z*n defined in

this way may not get closer to as *n *gets large.

Because of the possible spatial correlation in the

(2-8)

data, the size of the sampling region must be large

in relation to the correlation length in order for *Z*n

to accurately estimate .

may be obtained. This function has a key role in

(3) In addition to the mean of *Z(x)*, its varia-

geostatistical analyses. It is a measure of associ-

ation between values obtained at point *x*1 and those

bility or dispersion is also of interest, and this

obtained at point *x*2. If values at these two spatial

variability is most commonly measured by the

locations tend to be greater than average or less

squared deviations of *Z(x*) from ( *x*) and denoted

than average at the same time, then the covariance

by F2(x).

will be positive. However, if the values vary in the

opposite direction (that is, one tends to be larger

than average when the other is less than average,

F2 (*x*) = *+ *[ (*Z *(*x*) & (*x*))2]

(2-6)

and vice versa), the covariance will be negative.

The **(spatial) standard deviation ***F(x) *is the

(3) Because *C*(*x*1,*x*2) is an unknown population

square root of the variance. The following exam-

parameter, it too must be estimated using a sta-

ple illustrates the difference between the popula-

tistic computed from sample data. To make this

tion variance, which has been defined above, and a

possible, it is often assumed that the covariance

sample variance.

function depends only on the distance between

points, which is defined as the **lag ***h, *and not on

their relative location or orientation,

2-4