Geoscience Reference
In-Depth Information
determinations generally can be subdivided into a number of separate components.
In some applications the original data are stochastic in that they can be described by
random functions. However, often the main component of spatial variability is
deterministic, either because it is related to differences between rock units sepa-
rated by discontinuities (contacts), or because there are regional trends. The latter
can be extracted from the data by a variety of methods; e.g., by trend surface
powers of the inverse of distance, by various methods of kriging, by using splines,
or by means of other methods of signal extraction. After extraction of a determin-
istic component, the residuals generally are stochastic in that they can be described
by means of spatial random functions. In the simplest case, these residuals are
uncorrelated and their correlogram is a Dirac delta function representing white
noise. Measurement errors would create white noise. If extrapolation towards the
origin by means of a function fitted to the correlogram results in a variance that
significantly exceeds variance due to measurement errors, this would create a
pseudo-nugget effect hiding strong autocorrelation over short distances.
6.2.1 Finite or Infinite Variance?
A problem of considerable interest for spatial series (and for time series as well) is
whether or not the random variable used for the modeling is stationary in that it
would have a definite mean and finite variance. Stationarity implies intrinsic
stationarity but intrinsic stationarity does not imply stationarity. Jowett (
1955
)
was among the first to assume intrinsic stationarity rejecting the commonly made
automatic assumption of existence of a constant or variable mean
μ
. Matheron
(
1962
) initially introduced the variogram under the assumption of intrinsic
stationarity. His original approach recently was summarized by Serra (
2012
).
Box 6.5: Pseudo-Parabolic Behavior at the Origin
Like Matheron, Serra (
2012
, p. 59) defines the variogram as: 2
ʳ
(
h
)
¼
f
(
x
+
h
)]
2
where
E
[
f
(
x
)
(
h
) is the semivariogram as used in this chapter,
and
f
(
x
) denoted the value of a regionalized random variable at location
x
.
In an example, Serra uses a Poisson process with parameter
ʳ
to generate
values along a line. At every next equidistant location a value drawn from the
Poisson population is added to the value at the preceding location. This
artificial series does not have a finite variance but the increments have
variance equal to
ʻ
, and values for a finite segment of length
L
along
the series can be assumed to have finite variance
ʻ
2
. Suppose now that
˃
a
pseudo-covariance
is
estimated
in
the
usual
way
as:
(continued)
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