ETL 1110-1-175
30 Jun 97
(2) In example 1, no assumptions were needed
(1) If the scenario presented in example 1 is
again used, the sample variance Sn2(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
n
the mean will probably change depending on the
1
2
Sn
(x) =
sampling location. In addition, usually only one
n & 1 i=1
(2-7)
observation is available at any particular location.
Therefore some assumptions regarding the struc-
2
Zi (x) & Z n (x)
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
Zi(x) values from their sample mean . The sample
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 Z1(x), Z2(x), ..., Zn(x). However, in
a homogeneous aquifer, for example, might be
the limit as n increases, Sn2(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 Sn 2(x) is a random variable.
spatial locations x1, x2, ..., xn
(2) The mean and variance can both be calcu-
1
Zn =
Z (x1) % Z(x2)
lated from the probability distribution of Z(x).
n
(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(x1) and
Z(x2) the (spatial) covariance function,
However, in contrast to example 1, Zn defined in
this way may not get closer to as n gets large.
C (x1, x2) = +
Z (x1) & (x1)
Because of the possible spatial correlation in the
(2-8)
data, the size of the sampling region must be large
Z(x2) & (x2)
in relation to the correlation length in order for Zn
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 x1 and those
bility or dispersion is also of interest, and this
obtained at point x2. If values at these two spatial
variability is most commonly measured by the
(spatial) variance, defined to be the mean of
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(x1,x2) 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
c. Example 2.
their relative location or orientation,
2-4
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