locations, which is very important, because geo-

is used to denote the mean, or expected value, of

the bracketed term, in this case *Z*(*x*). It is intui-

statistics is based on using a measurement of a

regionalized variable at one location to gain infor-

tively helpful to think of the expectation as an

average. In fact, if the distribution of *Z*(*x*)

mation about values of the variable at another

location. The notion of distribution of *Z(x) *at a

assigned equal probability to a finite number of

values, then the expectation of *Z*(*x*) would indeed

single location is readily generalized to two or

more locations. For two locations, if we let *x*1 and

be the simple average of these numbers. In geo-

statistics, however, *Z*(*x*) is usually assumed to take

on any value in a continuous range of possible

values, rather than being limited to a discrete set of

values. In this case, calculus needs to be used to

that both *Z *(*x*1) # *c*1 and *Z *(*x*2) # *c*2. If the vari-

define the expectation. The following example

ables *Z*(*x*1) and *Z*(*x*2) are statistically independent

illustrates the difference between averages and

expectations.

of one another, then the joint probability distri-

bution can be obtained as the product of the indi-

vidual probability distributions,

(1) An experiment consists of injecting a con-

servative tracer at a particular well in a steady-

(2-1)

state groundwater flow system and measuring the

' *P *[*Z *(*x*1) # *c*1] *P *[*Z *(*x*2) # *c*2]

concentration, *Z*1(*x*), of the tracer in a neighboring

well 24 hr later. The tracer is then allowed to flush

However, in most applications, *Z*(*x*1) and *Z*(*x*2) will

from the system, and the experiment is repeated a

not be statistically independent and their joint

second time to obtain another concentration mea-

distribution cannot be obtained from the individual

surement, *Z*2(*x*), at the same location. If this

distributions. When this joint distribution descrip-

process is repeated *n *times, *n *concentration mea-

tion is applied to more than two locations, specifi-

surements *Z*1(*x*), *Z*2(*x*), ..., *Z*n(*x*) would be obtained,

cation of the full spatial distribution of *Z *would

all at location *x*. The average concentration level

require knowing the joint distribution of *Z*(*x*1), ...,

at location *x *is

1

with the full set of distribution functions of *Z(x) *is

(2-3)

not feasible and is not done.

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

(6) To simplify the problem even further, vari-

ous parameters of the distributions are usually

which would change depending on *n *and on the

considered rather than dealing with the entire dis-

actual values obtained for *Z*1(*x*), *Z*2(*x*), ..., *Z*n(*x*).

tributions. The parameter most commonly used to

However, in the limit as *n *increases, *Z*n (*x*)

characterize a distribution is the **mean**, or, because

becomes closer and closer to the true mean, or

the mean in geostatistical applications depends on

expected, concentration *(x*):

the spatial variable *x, *the mean may be called the

(2-4)

referred to as the expectation (*E*) of the random

variable *Z(x)*, and the symbol *m *is used in this

This theoretical limit is a constant value, or **popu-**

report to denote this expectation Thus,

(*x*) = *+ *[ *Z *(*x*) ]

(2-2)

random variable, or a property of the particular

sample that is taken.

2-3