ETL 1110-1-175
30 Jun 97
(6) To summarize, the basic model frame-
will adopt the variogram as the primary tool for
work that will be used throughout the ETL is the
analyzing spatial dependence in the remainder of
following: the value of a measurement z(x) (con-
this ETL.
centration, porosity, hydraulic head, and so on) at
b. As was the case with the covariance func-
location x of a two-dimensional region is the value
of a regionalized random variable, Z(x), with mean
tion, it is necessary to distinguish between the
( x) and stationary covariance function C(h,a).
theoretical variogram, which is a population
Other assumptions may be added in the applica-
parameter, and the sample variogram, which is an
tions sections to analyze specific data sets, but this
estimator of the theoretical variogram obtained
from observed data. The theoretical variogram
framework will be the basic framework from
of a regionalized random variable, ((x1 ,x2) is
which many of the results will be derived. In some
situations, the covariance stationarity assumption
defined as one half of the variance of the difference
between residuals at locations x1 and x2:
may be relaxed, for instance, when using the linear
variogram described in the next section.
1
((x1, x2) =
Var [Z ( (x1) & Z ((x2)]
(2-17)
2
2-3. Variograms
Because the residuals have been mean-centered, as
a. Regionalized random variables differ from
shown in Equation 2-16, they have a mean of zero.
classical (ordinary least-squares) regression
Therefore, using the well-known formula for the
models in that the residuals, defined as the devi-
variance of a random variable X
ations of the regionalized random variable from its
mean and denoted by
Var (X) = + X 2 & (+X)2
(2-18)
Z ( (x) = Z (x) & (x)
(2-16)
it is seen that Equation 2-17 is equivalent to
are related to one another, whereas the residuals in
1
( (x1, x2) =
+ [Z ( (x1) & Z ((x2)]2 (2-19)
a regression model are generally assumed to be
2
independent. Thus, in the regionalized random-
variable model, observed values of the residuals
The theoretical variogram is always non-negative,
from sampled locations contain valuable informa-
with a small value of g indicating that the residuals
tion when predicting the value of Z(x) at unsam-
at locations x1 and x2 tend to be close and a large
pled sites. The relationship among the residuals
value of 8 indicating that the residuals tend to be
can be understood by examining the variogram,
different. Equation 2-19 is sometimes called a
which is a tool that is widely used in geostatistics
for modeling the degree of spatial dependence in a
, but will be referred to in this ETL as a
regionalized random variable. Although the vario-
variogram.
gram is closely related to the covariance function,
there are some important differences between the
c. It would be ideal to know the theoretical
variogram and covariance function that will be
variogram before taking observations, but unfortu-
described below. The covariance function, and
nately, it must be estimated using sample data. To
related correlation function, are more commonly
facilitate variogram estimation, it is usually
used in basic statistics courses than the variogram,
assumed in a similar manner to the covariance
so many readers may be more familiar with the
function that ( depends only on the lag,
former concepts. However, the variogram is more
widely used in geostatistics, and because of this we
2-6
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