ETL 1110-1-175
30 Jun 97
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 x1 and
be the simple average of these numbers. In geo-
statistics, however, Z(x) is usually assumed to take
x2 be two distinct locations, then the joint proba-
bility distribution is defined to be the probability
on any value in a continuous range of possible
P [Z (x1) # c1, Z (x 2) # c2] for any constants c1 and
values, rather than being limited to a discrete set of
values. In this case, calculus needs to be used to
that both Z (x1) # c1 and Z (x2) # c2. If the vari-
define the expectation. The following example
ables Z(x1) and Z(x2) 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-
b. Example 1.
vidual probability distributions,
P [Z (x1) # c1, Z (x2) # c2]
(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 (x1) # c1] P [Z (x2) # c2]
concentration, Z1(x), of the tracer in a neighboring
well 24 hr later. The tracer is then allowed to flush
However, in most applications, Z(x1) and Z(x2) 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, Z2(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 Z1(x), Z2(x), ..., Zn(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(x1), ...,
at location x is
Z(xn) for any set of n spatial locations and for any
n; however, except in very special cases, working
1
Z n (x) '
Z1(x) % Z2 (x)
with the full set of distribution functions of Z(x) is
n
(2-3)
not feasible and is not done.
% ... % Zn (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 Z1(x), Z2(x), ..., Zn(x).
tributions. The parameter most commonly used to
However, in the limit as n increases, Zn (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
spatial mean, or the drift. In statistics, the mean is
(2-4)
Zn (x) 6 (x) as n increases
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,
lation parameter, as opposed to Zn(x) , which is a
(x) = + [ Z (x) ]
(2-2)
random variable, or a property of the particular
sample that is taken.
2-3
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